Figure 1. Simulation of an electron moving in free spac and then hitting a potential. The original FDTD-Q schem was employed with µ = 0.46875 and no **absorbing** **boundary** condition. Here, the horizontal coordinate is k and the ver- tical coordinate is ψ real .

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functions is employed to compute the Floquet **absorbing** **boundary** condition (FABC) formulation in the vector 3D finite element time domain (FETD) modelling of doubly periodic structures. This novel implementation (VF-RC-FETD-FABC) results in significantly lower computation time requirements than the standard convolution (SC) implementation. The numerical examples presented show reduction in computation time requirements by a factor of at least 3.5. In addition, a time window in excess of 50,000 time-steps is recorded over which practically stable results are obtained. This temporal window is sufficiently large to allow the modelling of many practical problems. Results from four such problems are presented confirming the accuracy and speed of the VF-RC-FETD-FABC software.

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Due to the limitation of the computer capacity, appropriate **boundary** condition should be assigned at the boundaries of the domain when we solve the electromagnetics problems with the finite difference time domain (FDTD) method. To solve this problem, many **absorbing** **boundary** condition (ABC) algorithms were proposed [1–5]. The presentation of Brenger’s perfectly matched layer (PML) ABC [6– 8] promoted the development of FDTD a big step. Because of its perfect absorption effect, it is widely used in numerical solution of EM scattering problems [9–15]. Afterward, scholars proposed many modified PML ABCs, such as uniaxial PML (UPML) [16, 17], convolutional PML (CPML) [18]. These algorithms all have better accuracy. Although there are so many better ABCs, Liao’s ABC

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The numerical simulation of wave propagation in the Earth is very useful for understanding geophysical phenomena such as earthquakes. It is also a noninvasive and efficient tool to explore limited regions of the subsurface. For instance, numerical waves are used by geophysicists to detect possible stocks of hydrocarbons in regions of the subsurface that are tricky to reach. In this case, it is necessary to couple the wave equations with **absorbing** **boundary** conditions (ABC). Another approach is to modify the wave equations within a layer surrounding the computational domain. It is the so-called PML (Perfectly Matched Layer) technique. The use of ABCs has been suggested long ago [9, 10] while the PMLs have emerged later [5, 6]. In the case of isotropic media, PMLs have clearly demonstrated their supremacy on the ABCs. They are easy to implement and do not generate spurious waves. To achieve the same level of accuracy with ABCs, higher order **boundary** conditions must be considered and difficulties of construction and implementation occur. Moreover, computational costs are significantly increased compared to those generated by PMLs. However, PMLs suffer from stability problems for different classes of anisotropic media [3], especially in TTI (Titled Transverse Isotropic) environments that are of great interest for oil exploration.

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For application to the simulation of the electromagnetic scattering and propagation in the free- space, the LOD-FDTD method should have an eﬃcient **absorbing** layer in the computation field. In [7, 8], the split-field perfectly matched layer (PML) and convolution PML have been applied to LOD-FDTD method. And the Mur’s and uniaxial anisotropic PML (UPML) **absorbing** **boundary** condition [9–12] have been implemented within the two-dimensional LOD-FDTD method [13]. More recently, there are some other PML-ABC applied in the LOD-FDTD method such as LOD-CPML [14] and LOD-SC- PML [15]. In this paper, the UPML-ABC is applied to the LOD5-FDTD method. Using the proposed UPML-ABC, the reflection error of an electric dipole source and target field phase distribution of a sinusoidal source are computed. The results of these simulation experiments show that the UPML- ABC can be used eﬃciently in the LOD5-FDTD method.

simulation of a tsunami with the LDW model is usually very expensive in terms of computational costs, which are roughly 30–100 times the costs of LLW tsunami simulations because of the iterative processes required to solve the finite difference equations. We note that because the PML condition is applied only once, during the last of the iterations (see Eq. 19), the additional cost of introducing the PML is negligibly small. Therefore, use of the PML **absorbing** **boundary** condition alterna- tive to extending the model area greatly reduces com- putational costs. The rapid, high-resolution simulation of tsunamis with robust **absorbing** **boundary** conditions for a model of moderate size should be useful for the repetitive simulation of tsunamis that is required, for example, for inversion studies of source fault ruptures and/or computation of the large number of Green’s functions (e.g., Baba and Cummins 2005; Tsushima et al. 2009) required for tsunami forecasting or source inversion.

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The treatment of **absorbing** **boundary** conditions at corners for differential-equation- based ABC’s is not a new problem. Earlier works were performed by Bamberger et al. [2] and Collino [10]. These earlier approaches formulate the corner absorbers by deriving corner compatibility conditions which are necessary to ensure the regularity of the solution of the problem when the data are smooth. In a more recent work [35], Vacus also describes an algebraic procedure for deriving corner compatibility conditions. It was recently highlighted by Hagstrom [22] that a more straightforward approach would be useful. Here, we present an approach that is simple and leads to an easy implementation of the corner **absorbing** boundaries. We will formulate the absorbers for non-orthogonal corners, with orthogonal corners being a special case.

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In a naive treatment of the unbound states, a scattering **boundary** condition should be imposed among the interacting particles, but imposing the scattering **boundary** condition for three particles is complicated in general. In particular, the exact treatment of the three-body (or few-body) scattering state requires special techniques for the long range interaction, such as the Coulomb force [2]. In order to avoid the complexity of the three-body scattering **boundary** condition, the scattering problem is often reduced to the bound-state-like problem by transforming the Hamiltonian of a total system into a non-Hermite form. There are two representative methods of the non-Hermitian transformations: the complex scaling method (CSM) [3–5] and the method of the **absorbing** **boundary** condition (ABC), or the method of the complex **absorbing** potential (CAP) [6–11].

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introduction of a radiation condition in any unbounded direction: waves should radiate outwards from a source toward an unbounded direction without any spurious wave motion in the reverse direction. Irregularities in the geometry of the domain or the physical material often require a numerical solution of the problem, thus requiring the use of a bounded domain, along with an artificial **boundary** condition named as **absorbing** **boundary** condition (ABC). These ABCs are expected to absorb outgoing waves and mimic the effect of the truncated unbounded part. Since the 1970s many researchers have proposed several ABCs, which are classified into two broad classes: differential-equation-based and material-based. Differential-equation-based ABCs are obtained by factoring the wave equation into outward and inward propagating operators and permitting only outgoing waves by eliminating the inward propagation operator. Material-based ABCs, on the other hand, are realized by surrounding the computational domain with a fictitious material that dampens the outgoing waves. Differential-equation-based ABCs can be further classified into two sub categories: exact (global) ABCs and approximate (local) ABCs. Computation with global **absorbing** boundaries involves obtaining the exterior Green’s function and coupling it with the interior, which involves expensive convolution operations [1]. Global ABCs are useful for small-scale wave propagation problems and lead to highly accurate results. In spite of many innovations, they still tend to be computationally expensive for large scale problems. With large-scale problems in mind, we focus our discussion on local ABCs and material based ABCs. In particular, we develop in this chapter an effective ABC for large-scale elastic wave propagation problems. Furthermore, we limit our discussion to polygonal domains with straight computational boundaries, and note that the use of circular and curved boundaries (see e.g. [2]) could be an efficient alternative to polygonal boundaries.

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Figure 1 shows the distri- bution of the energy eigen- values obtained by solv- ing Eq. (4) for the identi- cal three-bosons system un- der the **absorbing** bound- ary condition. If the ab- sorbing potential is switched oﬀ, the eigenvalues are lo- cated along the real (hor- izontal) axis. The energy eigenvalues are distributed in the complex energy plane, when the **absorbing** poten- tial is switched on. In the negative real energy region, there is a bound state (open square), which is almost in- variant under the **absorbing** **boundary**. In the positive energy region, the two reso- nant poles (solid circle and solid square) are separated from the continuum spectra (open circles).

In this thesis, we present a new direction for developing ABC’s for Molecular dynamics. The basic ideas stem from earlier research linking PML with local **absorbing** **boundary** conditions [12] to create a class of **boundary** conditions called Perfectly Matched Discrete Layers(PMDL)[13]. PMDL is essentially PML discretized using linear finite ele- ments with mid-point integration. It was shown that the integration error exactly cancels the discretization error, thus resulting in perfect matching even after the discretization of the exterior. Furthermore, it was shown that PMDL is equivalent to rational-approximation based ABC’s and thus inherits their efficiency while retaining the flexibility of PML. The main limitation, however, is that PMDL is perfectly matched with the continuous interior, but not with the discrete interior. Thus PMDL, like other ABC’s for wave equation, works well for low frequency limits, but fails in **absorbing** high frequency phonons.

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The plane-wave propagation in a material whose permittivity and permeability are assumed to be simultaneously negative is theoretically investigated by Veselago [1]. Recently, several papers have exposed the usefulness of double negative (DNG) medium with negative permittivity and permeability [2–5]. However, in order to further study unusual electromagnetic phenomena in DNG medium, full-wave numerical simulations have become more and more important. The FDTD method is a good choice for these electromagnetic problems. The **absorbing** **boundary** condition in FDTD is required to truncate the computation domain without reﬂection in the simulation of DNG medium properties and applications. An **absorbing** **boundary** condition for DNG medium has been proposed [6, 7]. Since ﬁrst introduced by Berenger in 1994 [8], the perfectly matched layer (PML) has become the most popular and eﬃcient **absorbing** **boundary** condition. Unfortunately, standard versions of PML are inherently unstable when they are extended to truncate the **boundary** of DNG medium without any modiﬁcation [9, 10]. Recently, a nearly PML **absorbing** **boundary** condition for DNG medium is discussed, and 50-cell layers for NIMPML are used to truncate the DNG medium [10]. Later, a modiﬁed PML **absorbing** **boundary** condition based on the complex- coordinate stretching variables has been proposed for PSTD method [11]. In this paper, a modiﬁed uniaxial PML (UPML) which is stable for the DNG medium is presented. It is worth noting that only 10- cell layers for UPML are used and provide a clearly reduced error to truncate the DNG medium.

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In [13] we presented a new method, which shares some features of both the PML and the high-order ABC, but enjoys some of the advantages that each of them lacks. In the new method, called the Double **Absorbing** **Boundary** (DAB) method, a high-order ABC is applied on two parallel artificial boundaries, which are a small distance apart. Auxiliary variables are defined on the two boundaries and in the thin layer beteween them. Like the PML, the DAB does not require special treatment of corners. The algebra involved is relatively simple, since no elimination of normal derivatives is needed. As in the method of high-order ABCs on a single **boundary**, DAB is clearly associated with the notion of convergence; one can approach the exact solution arbitrarily closely (up to discretization error) by increasing the order P, with only linearly-increasing cost. The numerical prop- erties of DAB, like accuracy, stability and sensitivity to discretization, are similar to those of a high-order ABC on a single **boundary**.

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Inspired by the blood vessel structures and microfluidic channels, bounded cylindrical envi- ronments have been considered for DMC systems. In [14], diffusion communication channels inside a microfluidic chip represented by a rectangular propagation environment with elastic (i.e., reflective) walls was characterized via particle-based simulation. In [15], a cylindrical DMC model with **absorbing** walls and no flow was considered. The hitting times and probabilities were obtained from simulation results. The response to a pulse of carriers, released by a mobile transmitter, was measured by receivers positioned over the vessel wall in [16], also based on simulation results. In [17], a cylindrical DMC environment was considered where the receiver partially covers the cross-section of a reflective cylinder. The distribution of hitting locations is again obtained from simulation results. The authors in [18] considered the diffusion in a cylinder with reflective walls and non-uniform fluid flow. Assuming a transmitter point source, the channel impulse responses for two simplifying flow regimes referred to as dispersion and flow-dominant, were derived. In [19], the authors obtained the channel impulse response for a 3-D microfluidic channel environment in the presence of flow where the boundaries are reflective. Also, in [20] we obtain the concentration Green’s function in a biological cylindrical environment where the **boundary** is covered by receptor proteins and information molecules are subject to both flow and chemical degradation.

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With the ultimate goal of devising effective **absorbing** **boundary** conditions (ABCs) for general anisotropic media, we investigate the well-posedness and accuracy aspects of local ABCs designed for the transient modeling of the scalar anisotropic wave equation. The ABC analyzed in this paper is the perfectly matched discrete layers (PMDL), a simple variant of perfectly matched layers (PML) that is also equivalent to rational approximation based ABCs. Specifically, we derive the necessary and sufficient condition for the well-posedness of the initial **boundary** value problem (IBVP) obtained by coupling an interior and a PMDL ABC. The derivation of the reflection coefficient presented in a companion paper (Chapter 2) has shown that PMDL can correctly identify and accurately absorb outgoing waves with opposing signs of group and phase velocities provided the PMDL layer lengths satisfy a certain bound. Utilizing the well-posedness theory developed by Kreiss for general hyperbolic IBVPs, and the well-posedness conditions for ABCs derived by Trefethen and Halpern for isotropic acoustics, we show that this bound on layer lengths also ensures well- posedness. The time discretized form of PMDL is also shown to be theoretically stable and some instability related to finite precision arithmetic is discussed.

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We also assume that the dielectric is backed by a supraconductive material with an infi- nite conductivity, so one side of the computational domain will have a perfectly reflecting **boundary** condition. Artificial boundaries on the other three sides are assumed, producing an approximating finite computational domain. Energy will also reflect off the boundaries on these other three sides of the computational domain, and there is a critical need to prevent or delay this energy’s return to the domain of interest. We use Perfectly Matched Layers to absorb incident waves without reflection and damp the absorbed waves, and we use a (partially) **absorbing** **boundary** condition to limit the amount of energy that reflects off a **boundary** of the computational domain. When the outgoing wave returns to the domain of interest after twice traversing the PMLs, the wave can be sufficiently attenuated to be negligible when compared to the reflections from the dielectric.

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The purpose of this work is to conduct an analytical study of the behavior of the high-order spurious modes induced by the new DtN2 **boundary** condition, and to establish a sharper estimate that provides a better understanding on their asymptotic behavior in the high-frequency regime. We prove that all the high-order reflected prolate spheroidal modes (whether they are propagating, evanescent, or grazing modes) decay — in fact — exponentially as ka tends to ∞. This result is very important to the performance of this **absorbing** **boundary** condition since it shows that the effect of these spurious waves on the accuracy level — if any — is negligible in the high frequency regime. Hence, this result provides practitioners with the needed confidence to employ the proposed **boundary** condition on artificial boundaries that are “close” to the considered scatterer’s **boundary**, leading therefore to small computational domains. Note that the situation is not the same in the case of the standard DtN2/BGT2 **boundary** conditions (recall that in 3D, the standard DtN2 and BGT2 conditions coincide [11]). More specifically, in the high-frequency regime, the high-order spurious modes (whether they are propagating, evanescent, or grazing modes) created by the standard BGT2/DtN2 **boundary** condition do not decay, as observed in [2], requiring therefore to place the artificial **boundary** very far from the obstacle to avoid the deterioration of the accuracy level due to the possible contamination of these non physical modes. Consequently, the present study demonstrates analytically the superiority of the new DtN2 **boundary** condition designed for prolate spheroidal boundaries over the standard BGT2/DTN2 **boundary** condition. Recall that previous numerical studies [3, 4, 17] have already indicated that the new DtN2 **boundary** condition outperforms the standard BGT2/DTN2 **boundary** condition. Hence, this study suggests DtN2 to be the primary **absorbing** **boundary** condition to be employed when solving high-frequency acoustic scattering problems by elongated scatterers. Note that the suggested **boundary** condition can also be used in the case of two-dimensional scattering problems by electromagnetic waves since such problems can be formulated using the Helmholtz equation, the Dirichlet (resp. Neumann) **boundary** condition on the scatterer(s) for the scattered field with E -polarization (resp. H-polarization), and the Sommerfeld condition [6].

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This thesis is about developing a conformal object using the finite difference time domain (FDTD) method. Theoretical analysis nowadays frequently uses to analyze problem regarding electromagnetic in computational algorithm. The FDTD method by numerically implemented and time based simulation results in better response of wide band frequency. In the FDTD there are several others principles that have been used in modelling the object; they are **absorbing** **boundary** condition (ABC), near Field Far Field (NFFF) and Scattering Parameter. This thesis also uses Radar Cross Section (RCS) characteristic to analyze the radiation pattern of the object at particular incident plane wave direction. The source of the plane wave was determined by the angle of phi and theta in the Matlab function. Once the simulation started, GUI immediately shows simulated object. After FDTD iteration completed the radiation pattern on the polar graph plotted in direction of xy-plane, xz-plane and yz-plane. This pattern can be used to analyze the object characteristic, regarding the frequency of the incident plane wave. From the radiation pattern, we can see that conformal (sphere) show less reflection than the cubical object as comparison.

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A pure FEM solution is utilised by Banimahd et al. [15] and Kouroussis et al. [16]. A challenge presented by this approach is that **absorbing** **boundary** conditions must be implemented at model boundaries to prevent spurious waves contaminating the solution. Kouroussis employed ABAQUS’s infinite element library to absorb such waves. Absorption performance is enhanced for excitations at the centre of the sphere by modelling the soil as a spherical domain. Despite this, the performance of **absorbing** **boundary** conditions decreases as the distance between excitation and **boundary** is reduced. Therefore when the excitation location deviates from the central position, performance degrades.

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As a highly efficient **absorbing** **boundary** condition, Perfectly Matched Layer (PML) has been widely used in Finite Difference Time Domain (FDTD) si- mulation of Ground Penetrating Radar (GPR) based on the first order elec- tromagnetic wave equation. However, the PML **boundary** condition is diffi- cult to apply in GPR Finite Element Time Domain (FETD) simulation based on the second order electromagnetic wave equation. This paper developed a non-split perfectly matched layer (NPML) **boundary** condition for GPR FETD simulation based on the second order electromagnetic wave equation. Taking two-dimensional TM wave equation as an example, the second order frequency domain equation of GPR was derived according to the definition of complex extending coordinate transformation. Then it transformed into time domain by means of auxiliary differential equation method, and its FETD equation is derived based on Galerkin method. On this basis, a GPR FETD forward program based on NPML **boundary** condition is developed. The me- rits of NPML **boundary** condition are certified by compared with wave field snapshots, signal and reflection errors of homogeneous medium model with split and non-split PML **boundary** conditions. The comparison demonstrated that the NPML algorithm can reduce memory occupation and improve cal- culation efficiency. Furthermore, numerical simulation of a complex model verifies the good absorption effects of the NPML **boundary** condition in complex structures.

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