The first step of the **boundary** **element** **method** (BEM) is the transformation of the physical problem at hand to an integral equation. The latter is defined frequently solely on the **boundary**, in which case the dimensionality of the problem is reduced by one. The presence of domain integrals in the BEM formulation implies domain discretization and this makes the BEM inefficient when compared with domain discretization techniques such as finite **element** **method** (FEM) or finite difference **method** (FDM). Thus, many efforts have been made to convert the domain integral into a **boundary** one (Marin et al. [20]; Wen and Khonsari [21]; Javaran et al. [22]). One of the most widely used techniques to accomplish this task is the **dual** **reciprocity** **boundary** **element** **method** (DRBEM) developed by Nardini and Brebbia [23] in the context of two- dimensional (2D) elastodynamics and has been extended to deal with a variety of problems wherein the domain integral may account for linear-nonlinear static-dynamic effects (Brebbia et al. [24]; Wrobel and Brebbia [25]; Partridge and Brebbia [26]; Partridge and Wrobel [27]; Partridge et al. [28]; El-Naggar et al. [29, 30]; Gaul et al.[31]; Fahmy [32-34]).

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A numerical model based on the **dual** **reciprocity** **boundary** **element** **method** (DRBEM) is extended to study the generalized magneto-thermo-viscoelastic transient response of rotating thick strip of functionally graded material (FGM) in the context of the Green and Naghdi theory of type III. The material properties of the strip have a gradient in the thickness direction and are anisotropic in the plane of the strip. An implicit-implicit staggered strategy was developed and implemented for use with the DRBEM to obtain the solution for the displacement and temperature fields. The accuracy of the proposed **method** was examined and confirmed by comparing to the obtained results with those known before. In the case of plane deformation, a numerical scheme for the implementation of the **method** is presented and the numerical computations are presented graphically to show the effect of the rotation on the temperature and displacement components.

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For two-dimension form, some of the recent computational ones include the application of a fully implicit finite-difference (see [23]), by Adomain-Pade technique (see [24]), by adopting variational iteration **method** (see [25]), by applying a meshfree technique (see [26] and [27]). Even more recently, more numerical works have been successfully carried out and nicely documented in literature; the application of the **dual** **reciprocity** **boundary** **element** **method** (see [28] and [29]), POD/discrete empirical interpolation **method** (DEIM)-reduced order model (ROM) (see [30]), and a combination between the homotopy analysis **method** and finite differences (see [31]).

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2) The difference between computation and measuring for QingYuang, KangLe, SangShui and LuoDong was large. The reason is another HVDC system (TianGuang) was then under bipolar operation mode with unbalanced power of 400MW, which influence the earth potential of the substations on the path of QingYuan-KangLe-Sang- Shui-LuoDong, when the measuring data were recorded. The event leads to the measuring data stand for not only the influence of XinAn HVDC system and the error. But however the correction of the **method** presented in this paper do not be influenced.

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The Effective Index **method** was first proposed by Knox and Toulios in 1970 (Knox and Toulois 1970) as an extension to the Marcatili’s **method** (Marcatili 1969) for the fundamental mode of a simple rectangular core waveguide. This resulted in the effective index **method** becoming one of the most popular methods in the 1970s for the analysis of optical waveguides whereby the rectangular structure is replaced by an equivalent slab with an effective refractive index obtained from another slab. The rectangular dielectric waveguide is divided into two slab waveguides in each transverse direction. The initial step solves the transcendental equation for a vertical slab waveguide by applying the appropriate **boundary** conditions. The effective index calculated in this step is then used as the refractive index of the horizontal slab waveguide and by solving the eigenvalue equation gives a good approximation to the effective index of the original waveguide structure. This **method** is significantly more efficient than those methods that solve the rectangular structure directly since only the solutions for slab waveguides are required. The advantage of the effective index **method** is that it can be applied to a wide variety of structures including channel waveguides, strip waveguides and arrays of such waveguides (Chiang et al. 1996) and also for various types of optical fibres and fibre devices (Chiang 1986b; Van de Velde et al. 1988). The disadvantage of this **method** is that it does not give good results when the structure operates near cut-off region. However, the simplicity and speed of the **method** have encouraged many engineers to search for different approaches that will improve the accuracy of the effective index **method** which subsequently lead to many different variants of the effective index **method** to be developed including the effective index **method** based on linear combinations of solutions (Chiang 1986a; Van Der Tol and Baken 1988) or the effective index **method** with perturbation correction (Chiang et al. 1996).

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RMSE are 0.01 and 0.016 for the period between 1960 and 2003, as shown in Figure 2 (left panel), which also demonstrates that relative deviation between the measured and predicted cumulative curves, i.e. the difference divided by the attained level of cumulative inflation, gradually decreases (right panel). Both differences in the left panel of Figure 2 are integrated of order 0, as the Phillips-Perron (PP) and the Dickey-Fuller (DF) tests show. Specifically, the PP test gives z( ρ )=-50.6 and -19.9 for the dynamic and cumulative series with the 1% critical value of -18.8; and z(t)=-6.96 and -3.54 with the 1% critical value of -3.63 (5% critical value is -2.95). The DF test gives z(t)=-6.97 and -3.56 with the 1% critical value of -3.63. Therefore, one can reject the null the series contain a unit root, i.e. both differences are stationary processes. The difference between the cumulative curves is an I(0) process and is a linear combination of two I(2) processes! This is the expression of the power of the **boundary** **element** **method** in science – integral solutions, when exist, suppress noise very effectively by destructive interference.

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In this work an improved numerical solution of the singular **boundary** integral equation of the 2D compressible ﬂuid ﬂow around obstacles is obtained by a **boundary** **element** **method** based on modiﬁed shape functions and cubic **boundary** elements. The singular **boundary** integral equation with sources distribution is considered in this paper, and for its discretization cubic **boundary** elements are used. The integrals of singular kernels are evaluated using modiﬁed shape functions which are deduced by using series expansions for the basis functions we choose for the local approximation models. A computer code is made using Mathcad programming language and, based on it, some particular cases are solved. In order to validate the proposed **method**, comparisons between numerical solutions and exact ones are performed for the considered test problems. The advantage of using modiﬁed shape functions for evaluating the singularities is pointed out through a comparison study between the numerical solution obtained by the **method** proposed in this paper and the one obtained by using a truncation **method** for evaluating the singularities. MSC: 65N38; 76M15; 76G25; 35Q35

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A parallel vortex-**boundary** **element** **method** has been developed for the simulation of three-dimen- sional wall-bounded ow on the Thinking Machines CM5. Various parallelization paradigms have been tested for the direct evaluation of the vortical and potential velocities (and their gradients,) and the fastest approach implemented in the code. Both vortical and potential ow evaluations displayed good parallelization eciency and acceptable speed-up over the Cray C90. Consequently, vortex- **boundary** **element** simulation of three-dimensional ow involving up to 100000 vortex elements is now feasible. The functionality of the code has been ascertained by the test problem of ow over an idealized trailer truck near the ground level at R e = 500. A total of 45000 vortex elements and 4176

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The focus of this work has been to obtain accurate tangential velocities on the **boundary**. The tangential ve- locity is crucial to predicting flow stability and is the basis for vorticity creation in some formulations. Er- rors can arise from a variety of sources. Methods based on solving the differential equation Eq. (3) yield tangential velocity errors as a result of approximate derivatives on the **boundary**. **Boundary** **element** solu- tions to Eq. (3) have superior convergence properties but suffer from the same difficulties when applying gradient operators (discrete and analytical) to the potential function. Large errors in the tangential velocity on the **boundary** can also occur if spatial variations in the normal velocity **boundary** condition are not accu- rately resolved.

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This paper presents the **boundary** **element** formulation of elastoplastic analysis of Reissner plates. The formulation follows closely the work by Karam and Telles (1998); however not only the plastic strain due to bending but also the plastic strain due to membrane are considered. The total incremental technique is applied in dealing with nonlinear system of equation. The cell discretization **method** using 9-nodes quadrilateral cell is employed to evaluate the domain integrals appearing in the formulation. Elastic-perfectly plastic material is considered. Throughout this paper, the cartesian tensor notation is used, with Greek indices varying from 1 to 2 and the Latin indices varying from 1 to 3.

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Since p ∈ H 1 ( Ω ) but p ̸∈ H 2 ( Ω ) , we can assume that p exists in an interpolation space, somewhere between H 1 and H 2 . Boffi et al. [12] stated that p ∈ H 3/2− ε ( Ω ), ε > 0, and they attributed their sub-optimal orders of convergence to the pressure being from this interpolation space. Unfortunately, as far as we are aware, error estimates for the spectral approximation of a function from an interpolation space such as H 3 / 2− ε do not exist. Thus, we can infer that the reason for the superior rates of convergence given above are the same as discussed in Section 7.1. For the original immersed **boundary** **method**, it has been shown in the literature [36,37] that second order convergence of the velocity with respect to mesh width (h-type) is obtained for sufficiently smooth problems. In this section, we have illustrated that higher rates of convergence can be obtained when using SE-IBM. Again, we emphasise that the majority of the error in the IBM originates in the spreading and interpolation phases; specifically, for the original IBM the largest error can be found in the approximation of the delta function. Given that the delta function is not a function, it makes more sense to approximate it distributionally (i.e. approximate its action on another function). This is the procedure employed in the IFEM [9] and the FE-IBM [10]. In the FE-IBM, and hence the SE-IBM, the action of the delta function is included through its sifting property. Polynomial interpolation is then used to construct the spreading and interpolation phases, as shown in Section 3. Therefore, as mentioned earlier, the main source of error comes from interpolation error, which is only dependent on the regularity of the function being interpolated. Thus, a function u ∈ H 2 should yield second order convergence, whilst a function u ∈ H 3 should yield third order convergence (assuming no impairment occurs from elsewhere).

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This paper is organized as follows: Section 2 introduces the theoretical background of this paper. In 2.1 an abridged formulation of SBFEM theory is treated. The standard pro- cedure for calculating SIFs is reviewed in 2.2. The novel proposed enhancements to the SIF calculation procedure, as well as the accompanying error estimator, is presented in 2.3. The newly proposed Hamiltonian Schur decomposition (HSchur), advocated for adoption in SBFEM, is treated in 2.4. Four applications are demonstrated in section 3. In the first numer- ical example (3.1), the issues associated with the current solution procedure in SBFEM are discussed, while demonstrating the ability of the proposed HSchur decomposition to over- come these, based on the example of an edge cracked square plate subjected to bending. The second numerical example (3.2), provides an exact solution for displacements, stresses and SIFs, establishing a new benchmark. The convergence behavior of SBFEM in terms of dis- placements, stresses and SIFs is discussed, offering a unique insight into the **method**. In the third numerical example (3.3), SBFEM is contrasted to XFEM and the standard FEM, based on floating point operation counts. This type of extensive contrasting to XFEM in terms of performance and computational complexity has not so far been offered in literature. The fourth numerical example (3.4), a slant crack in a finite body, investigates the performance of SBFEM in the worst case scenario and evaluates how it performs compared to XFEM and the standard FE-based approach. Section 4 summarizes the findings and provides direction for future research.

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after one step of subdivision. Thus, a simple remedy is to shrink all 1-form coefficients associated with shrunk edges by the same ratio. A more uniform vector field is achieved using this **method** (Fig. 5.7). From results, we notice that the enlarging phenomenon of vector fields in **boundary** cells are removed. However, the projection-scaling **method** has two drawbacks. First, in the example above we can see that streamlines in **boundary** cells exhibit some curved features because subdivision results are affected by initial **boundary** edges and corner vertices. Second, shrinking the form coefficient for a constant ratio is ad hoc and not robust enough for general meshes with more exotic **boundary** shapes.

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attainment and accuracy of the solution. To demonstrate that the improved performance of the BEM is owing to the use of the IRBFN interpolations to represent the variations of functions (velocity, traction, temperature, heat ﬂux and geometry) along the bound- ary, all other aspects of the analysis are kept the same, i.e. a single domain, the use of the Stokeslet fundamental solution (the primitive variables) and the standard treatments for the convective terms by a successive substitution scheme and linear triangular cell approximations. The present IRBFN-BEM can achieve a high Rayleigh number value of 1.0e7 using a relatively coarse and uniform mesh of 31 × 31 in the case of a square cavity, and 5.0e4 using a uniform mesh of 11 × 21 in the case of a horizontal concentric annulus. For the former problem, convergence was observed to be very slow for Rayleigh number above 1.0e7 which is here considered as a limit of the present approach. The remainder of the paper is organised as follows. In section 2, the governing diﬀerential equations of natural convection problem and the corresponding **boundary** integral formulations are summarised. A brief review of the indirect RBF networks is given in section 3. The pro- posed IRBFN-BEM scheme for the analysis of natural convection is presented in section 4. Sections 5 and 6 are to verify the validity of the present **method** through the simulation of natural convection ﬂow for a wide range of Rayleigh numbers in a square cavity and in a horizontal concentric annulus respectively. Section 7 gives some concluding remarks.

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Abstract— In this work a **boundary** **element** formulation for the analysis of plate-beam interaction is used in the analysis of practical building slabs and waffle slab. This formulation uses a **boundary** **element** with three degrees of freedom per node and the beam **element** is replaced by their actions on the plate, that is, a distributed load and end of **element** forces. From the solution of the differential equation of a beam with linearly distributed load the plate-beam interaction tractions can be written as a function of the nodal values of the beam. With this transformation a final system of equation in the nodal values of displacements of plate **boundary** and beam nodes is obtained and from it, all unknowns of the plate-beam system are obtained. The results show an excellent agreement with those from the a finite **element** analysis.

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Although full-domain IRBF methods are highly ﬂexible and exhibit high order convergence rates in their basic implementation, the associated fully-populated matrix systems can lead to poor numerical conditioning as the scale of a problem increases (Mai-Duy et al., 2008). The problem becomes critical with increasingly large data sets. Many techniques have been developed to reduce the eﬀect of the problem, including DD methods (Ingber et al., 2004; Tran et al., 2009), adaptive selection of data centres (Ling et al., 2006), RBF preconditioners (Brown, 2005) and RBF based compact local stencil methods (Mai-Duy et al., 2011). While a reliable **method** of controlling numerical ill-conditioning and particularly compu- tational cost, as problem scale increases, can be based on DD **method**, the use of compact local approximations facilitates the solution of a diﬀerential equation without having to deal with large systems of global equations. In this chapter, a parallel algorithm based on compact local integrated RBF and DD techniques is developed for the solution of **boundary** value problems (BVP). A large problem is ﬁrstly decomposed into many smaller problems, each of which is analysed in parallel, and secondly the acceleration of the convergence rate within each sub- region using groups of CLIRBF stencils is carried out by parallel CPUs. For the ease of presentation, in this chapter the terms sub-domain and process are used interchangeable.

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on the numerical solution as represented by the Courant number. This is deliberate. Our major concern here is to see how the models cope with highly convected flows. The Courant number is fixed (Cr = 0.2), the Peclet num- ber is increased from Pe = 10 to Pe = 100. Figure 5(a) represents the case where Pe = 10 and Cr = 0.2. Two dis- tinguishing characteristics are observed namely: there are error-free regions and regions characterized by spikes in absolute error values at the upstream and downstream ends of the problem domain. While the three models are able to accurately represent gradient-free regions of the problem specification, the spikes quantify attempts by the models to handle high gradient areas. While MOD 1 and MOD 3 values of absolute errors are almost identical, MOD 2 can be said to have performed moderately better. When Pe = 100, Cr = 0.2 (Figure 5(b)), the numerical schemes exhibit a trend similar to that of Figure 5(a) except that the spikes are now more pronounced. Convection is more pronounced as the governing equation now tends to be more hyperbolic. It is worthwhile to note the effects of the shockwave propagation and its interaction with the problem **boundary** conditions. As the shock wave approaches the outflow **boundary**, the influence of the inflow **boundary** becomes less significant. On the other hand, when the shockwave encounters the outflow, the zero outflow Dirichlet **boundary** condition is enforced and the wave recedes upstream. As a result of this interesting physics the numerical error increases dramatically away from the outflow and moves closer to the inflow region. We may mention that the physics becomes significantly different if the problem had admitted infinite boundaries with Neumann **boundary** condi- tions or if we had run the problem for long time periods.

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Figure 4 shows two different meshes. The first mesh was used for the implementation of the **method** of **boundary** integral equations. The second mesh was used for calculations using the finite **element** **method**. Figure 5 shows a comparison of the displacement results obtained with the two methods in the neighborhood of the fissure point. There is a good agreement between the two methods. The **method** involving the **boundary** integral equations has an immediate advantage compared to the finite **element** **method**. This advantage is at the mesh and tedious during the implementation of the finite **element** **method**. In the finite **element** **method**, the mesh refinement near the fissure point has a significant cost in the mesh discretization. However, despite this advantage, we note that when implementing the **boundary** integral **method** special attention must be paid in order to overcome the difficulties related to the presence of singularities in the expressions of displacement fields and tension, as well as the elimination of tension duplication at the singular points. The counting of the mesh edges and the **method** of **boundary** integral equations to identify tension duplication necessitated the use of software for the elimination of non-useful points and the non-square transformation matrices [A] and [B] in order to achieve a final linear system solution that could not be solved using the classical Gaussian **method**. A comparison of the displacements obtained by the two methods (i.e., the **boundary** integral equation **method** and the finite **element** **method**) validates our results for back meadows stresses at the crack tip and also validates our calculations of the stress intensity factors. These factors can be used to evaluate the material out of the pins. The small-scale examination of the

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Electromagnetic field is an advanced topic which requires thinking in abstract terms and with imagination. This paper presents authors’ experience in teaching electromagnetics at Tianjin University, China and demonstrates that this difficult subject can be tackled with the help of advanced numerical simulations and virtual experiments. Commercial simulation packages and in house developed software such as the Finite Elements **Method** (FEM), the **Boundary** **Element** **Method** (BEM) and an analytical **method** were used for this purpose. Overall, students have provided positive feedback for this teaching methodology.

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