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        Dynamic Adaptive Finite Element Analysis of Acoustic Wave Propagation Due to Underwater Explosion for Fluid-structure Interaction Problems

        2015-07-30 09:58:10SeyedShahabEmamzadehMohammadTaghiAhmadiSoheilMohammadiandMasoudBiglarkhani

        Seyed Shahab Emamzadeh, Mohammad Taghi Ahmadi, Soheil Mohammadi and Masoud Biglarkhani

        1. Islamic Azad University, Kangan Branch, Kangan 7557146845, Iran

        2. Department of Civil Engineering, Tarbiat Modares University, Tehran 14115-143, Iran

        3. School of Civil Engineering, University of Tehran, Tehran 4563-11155, Iran

        4. Department of Civil Engineering, University of Hormozgan, Hormozgan 3995, Iran

        1 Introduction1

        Most problems of dynamic fluid-structure interaction (FSI)in acoustic media around hydraulics and marine structures are solved by the finite element method (FEM). For example,dynamical response of a concrete dam subjected to underwater contact explosion has numerically investigated by Yu (2009) or ship shock modeling under far-field underwater explosion by Shin (2004). Although experimental works have down in this field such as Ghanaatet al. (1992). In his work, an experimental study of dam-water-foundation interaction has down on a full scale dam in China. However, even after more than 40 years of development of FEM, the question of estimating and controlling the discretization errors has remained as a major challenge. Several FEM modifications have been reported for improving the results. Sprague and Geers (2008) have developed a Legendre spectral finite elements algorithm for structural dynamics analysis. Rosset al.(2008; 2009) proposed the localized Lagrange multiplier method for acoustic FSI. In these coupled Euler-Lagrange works, fixed uniform meshes are considered for the whole analyses. As fine meshes are required where large gradients exist in the solution, such computations are not optimum.Other methods such as arbitrary Lagrangian-Eulerian (ALE)ones have been reviewed by Mair (1999). Kim and Shin (2008)have recently applied ALE techniques to underwater explosion analysis of a submarine liquefied oxygen tank. Lagrangian motion was computed at every time step, followed by a remap phase in which the spatial mesh was alternatively not re-zoned (as Lagrangian), re-zoned to its original shape (as Eulerian) or re-zoned to some more “advantageous” shape (as somehow between Lagrangian and Eulerian). ALE technique is usually employed to preserve a uniform mesh and not used to enhance the physical phenomena itself. Thus, its spatial description of the mesh is neither restricted to follow the material motions (as in Lagrangian) nor does it remain as fixed in space (as in Eulerian). One of the main concerns in acoustic finite element analysis is the adequacy of the finite element mesh to solve the dilatational wave equation that governs the fluid behavior. The Galerkin method provides good accuracy as long as the mesh is fine enough to comply with the maximum wave number. This is a criterion often too expensive even for moderate wave numbers. Moreover, a non-uniform finite element mesh is needed in many practical problems because increasingly finer grids are required near singularities and non-smooth boundaries. Nevertheless,such a discretization process is still unable to provide a proper resolution and order of the approximation at the required locations despite quite fine meshes. An h-adaptive finite element strategy for acoustic problems was presented by Bausys and Wiberg (1999), among the others. Their method’s key features are error estimation, adaptive mesh generation and re-meshing for finite element analysis using the superconvergent patch recovery technique for prime variables.In this process a highly reliable estimation of the discretization errors is crucial. Bouillardet al. (1996) implemented the original superconvergent patch recovery (SPR) technique for acoustic finite element analysis. They extended the original concept to complex variables and studied the reliability of the error estimation process.

        Tetambe and Rajakumar (1996) presented the error estimation strategy for acoustic analysis based on nodal averaging technique. A residual-based a posteriori error estimator for Helmholtz equation was presented by Harariet al.(1996). Recently, many efforts have been made for dynamic adaptive finite elements. Kadioglu and Sussman (2008)presented adaptive solution techniques for simulating underwater explosions and implosions. They solved several test problems to show the performance of their methods and validated the results by comparing shock speed, shock amplitude, and material interface speed, with benchmark results produced by Wardlaw (1998). Finally, their effort indicated that for some specific cases, e.g. investigating bubble growth and collapse dynamics, that the semi-implicit approach is significantly more efficient than an explicit approach.

        Early, in order to solving hyperbolic conservation laws adaptive mesh refinement techniques are developed (Bellet al.,1994). These extended to solve the compressible Navier–Stokes equations accordingly Skamarock and Klemp (1993)and Steinthorssonet al. (1995). The significant efforts have been performed by them in order to solve incompressible or weakly compressible flows adaptively by Stevens and Bretherton (1996), Howell and Bell (1997) and Pemberet al.(1998).

        Recently, an adaptive method is developed for solving one-dimensional systems of hyperbolic conservation laws,which combines the rezoning approach with the finite volume weighted essentially non-oscillatory scheme (Huaet al., 2015).They found this adaptive method exhibits more accurate resolution of discontinuities for a similar level of computational time comparing with that on a uniform mesh.

        Dapognyet al. (2014) propose a method for dealing with the problem of mesh deformation (or mesh evolution) in the context of free and moving boundary problems, in three space dimensions. Their method would have been considered in the fields of mesh generation, shape optimization, and computational fluid dynamics.

        Berrone and Marro (2009) presented space–time adaptive simulations for unsteady Navier–Stokes problems. The fluid domain was discretized by structured triangular finite element mesh and the unsteady flow was traced with a posteriori estimates and adaptive algorithms.

        In the present research an adaptive FE strategy has used that employ refined mesh only in wave front and provide reduced equations. This refining has also been a terminated optimal refining. Early works with traditional FE employ refined mesh in whole domain of analysis. The main difference of the content and advantages of the following is adapting the mesh refining with wave length and wave frequency. Usually in wave front the wave frequency is high and consequently wave length decreases, so a refining mesh has developed for optimal covering of wave front length.

        In the present study, the error estimation has been performed using curvature based error indicators for prime variables of finite element approximation. For mesh generation and re-meshing, the program MESH2D is adopted employing the advancing front technique. This program was originally prepared by Gilardo (1995) for compressible flow. The finite element code ZFEAP, prepared by Emamzadeh (2008), is used for adaptive fluid-structure interaction problem. Numerical examples are shown to illustrate the efficiency of the proposed error indicator and adaptive strategy procedure.

        2 Underwater explosion and pressure wave distribution

        The pressure load acting on a structure due to an underwater explosion (UNDEX) changes with respect to both time and space. The pressure time history at the standoff point (the point where the wave hits the structure first) is given. The incident pressurepIat a pointjwith vector coordinatexjcan be written as:

        where Pt (t)is the pressure time history at the standoff pointx0, andPx(xj) is the spatial function, at an arbitrary pointxj.For a plane wave:

        By considering the time delay required for the wave to travel from the standoff point to an arbitrary point, it is found that

        where

        wherexsis considered to be the specified source point of explosion.cfis the wave velocity in the fluid andτjknown as the “retarded time” corresponds to the time lag for the pressure wave to travel from the standoff point. Detailed basic formulation of the incident wave can be found in Cole (1948).

        3 Modeling of fluid-structure interaction

        phenomenon by CEL approach

        If a continuum deforms or flows, the position of the small volumetric elements changes with time. These positions can be described as functions of time in two ways. Lagrangian describes it as the movement of the continuum is specified as a function of its initial coordinates and time. Eulerian describes it as the movement of the continuum is specified as a function of its instantaneous position and time. In simulations with Lagrangian formulation the interface between two parts is precisely defined and tracked. In these simulations large deformation of a part leads to hopeless mesh and element distortion. In Eulerian analysis an Eulerian reference mesh, which remains undistorted and does not move is needed to trace the motion of the particles. The advantage of an Eulerian formulation is that no element distortions will occur. Disadvantageously, the interface between two parts cannot be described as precise as if a Lagrangian formulation is used. The ALE method can be considered a superset of both the Eulerian and Lagrangian method, since both types of mesh motions are incorporated within an ALE Scheme. The ALE method cannot be considered a superset for allowing an Eulerian region to interact with a Lagrangian interface. In the following,Lagrangian approach has been selected for structure and Eulerian approach has been selected for surrounding fluid.When a structure is exposed to UNDEX, it deforms and displaces the surrounding fluid, by the scattered pressure wave. Thus, the sum of the known incident pressure (as in Eq.(1)) and the unknown scattered pressure are applied to the structure as a result of the fluid-structure interaction. The equilibrium equation for small motions of an acoustic fluid with velocity-dependent losses is taken to be as:

        wherepis hydrodynamic pressure in excess to hydrostatic pressure.are velocity and acceleration vectors of fluid particles, respectively.ρfis fluid density andγis the“volumetric drag” (force per unit volume for unit velocity).Fluid behavior is assumed to be in viscid, linear, and compressible, so

        whereKfandfuare the bulk modulus and displacement vector of fluid particles, respectively. By dividing Eq. (5) byρf, taking its divergence, neglecting spatial variation of,and combining the result with the time derivatives of Eq. (6) one obtains the equation of transient motion for the fluid in terms of the fluid pressure:

        An equivalent weak form of Eq. (7) is obtained by introducing an arbitrary variational field,pd, and integrating over the fluid domainVf

        Integration by parts allows this to be rewritten as:

        Assuming thatpis prescribed onSfp, the equilibrium equation,Eq. (5), is used on the remainder of the boundary to relate the pressure gradient to the motion of the boundary:

        wherenis the inward unit vector normal to the fluid boundary.Using this equation, the termn×p? is eliminated from Eq. (9)to produce

        where, for convenience, the boundary “traction” term is defined as:

        In the absence of volumetric drag this boundary traction is equal to the inward acceleration of the particles of the acoustic medium, i.e.;

        Many engineering problems dealing with waves involve infinite domains. Usually, the infinite domain is truncated for computational purposes and the wave problem is solved in a finite domain. Non-reflecting boundaries (NRBs) have to be considered, which must allow the waves to leave the truncated domain avoiding spurious reflections that may pollute the solution in the interior of the computational domain of interest.

        There are many types of NRBs, which can be classified into two groups, namely, Non-Reflecting Boundary Conditions (NRBCs) and Non-Reflecting Boundary Layers(NRBLs). NRBCs are boundary conditions on the artificial boundary that absorb impinging waves. On the other hand,NRBLs have the property of absorbing waves that are traveling inside the layer.

        Finite element simulation of the time-dependent wave propagation in infinite media requires enforcing the transmitting boundary to replace the truncated far-field infinite domain so as to model the effect of the wave radiation towards infinity.

        The well-known Helmholtz equation governing the pressurep:

        where

        it is considered to denote the speed of sound in the fluid.

        If we consider only variations inx(the horizontal direction)we know that the general solution of Eq. (14) can be written as

        where two wavesFandGtravel in positive and negative directions ofx, respectively. The absence of the incoming waveGmeans that on infinite boundary we have only

        Thus

        whereF¢ denotes the derivative ofFwith respect to (x?cft)by eliminating the unknown functionF¢ the Sommerfeld formulation obtained.

        In Eulerian approach, the non-reflecting infinite boundarySfi, Sommerfeld equation (Sommerfeld, 1949) has been used as:

        For a rigid boundary,

        For the structural interface boundary,

        whereanis considered to be the normal acceleration.Application of the standard Galerkin discretization to the weak form of the governing equations of structure and fluid leads to a classical coupled problem, expressed by two set of second order differential equations as;

        whereM,DandKare the structural mass, damping and stiffness matrices. AlsoE,AandHare the fluid equivalent mass, damping and stiffness matrices, respectively. In the above equations pressure is defined as:

        wherep,pIandpscorrespond to the total, incident and scattered pressure waves respectively. By substituting Eq.(24) in Eq. (25), the fluid equation is obtained in terms of the unknown scattered pressure term, depicted byphereinafter for brevity. The resulting equation is solved together with Eq.(23) to obtain the response of the structure. The unknown functions are discretized as:

        whereBis the material elasticity matrix of the structure,Dis Rayleigh damping matrix,αandβare proportional to significant frequencies of structural response. The coupling matrixQrelates the structural and fluid nodal forces on interacting surfaces. The structure loading is due to an explosion occurring at a source point in the fluid. For fluid domain, similar matrices are defined as:

        where theKfis the bulk modulus of fluid.

        For solving the dynamic equations, Newmark implicit method with fixed time increment is applied in a staggered algorithm. In the following problems, a reliable time increment about 0.1 microseconds is adopted for time discreization.

        In the staggered algorithm based on the previous scattered pressure at each time step the structural response is calculated from which the new acceleration could in turn be exerted to the fluid domain boundary. This alternative domains calculation is followed iteratively in the same time step until a convergence be achieved.

        4 Adaptive finite elements

        In structural problems it is generally desirable to obtain a solution in which an energy norm of error is equally distributed within all elements. Such a norm of error can also be extended to viscous flow especially when it is relatively slow and nearly elliptic. However, according to Zienkiewiczet al. (2005) the energy norm has little significance at high speeds, a situation we face due to explosion, and thus we revert to other considerations which simply give an error indicator rather than an error estimator. Among the two available procedures such as gradient and curvature, the curvature based refinement will be adopted in this study.

        4.1 Curvature based refinement

        The error indicator should remain constant in each element.Theh-refinement process is applied to first-order triangular elements. Fig. 1 determination of error indicators over the elements is carried out by interpolation error approach. Ifx' is the local coordinate inside an element of length handpis a scalar function, the error inpis of order O(h2) (Peraireet al.,1987)

        wherephis the finite element solution andis a constant.

        Fig. 1 An element based error defined by Zienkiewicz et al.(2005)

        If, for instance, it is assumed thatp=phis at the nodes, i.e.the nodal error is zero, thenrepresents the values on a. The assumption that the nodal values of the functionpare exact is true only for certain types of interpolating functions and equations.However, according to Zienkiewiczet al. (2005), the nodal values remain always more accurate than elsewhere. An element subdivision is sought for equal distribution of errors.can be interpreted as a permissible error and so it can be simply insisted that whereep=is considered to be the user-specified error limit.

        If the shape functions ofpare assumed to be linear, then the second derivatives are difficult to determine. They are difficult to determine because they are clearly zero inside the element and infinity at the element interfaces. Some averaging processes have to be used in order to determine the curvatures from nodal computed values. In two and three dimensional problems, the second derivatives (or curvatures)tensors are given as:

        This requires determination of the principal values and directions. Determination of the second derivatives ofphneeds future elaboration. Despite linear elements the curvatures ofphshould be interpolated and a second-order polynomial has to be adopted over a local patch of linear elements.

        Such a polynomial can be applied in a least square manner to fit the values at all nodal points within a patch of elements sharing a particular node. In problems where the gradient of a function may be preferred to the curvature, the maximum value of the gradient ofp, for instance, can be easily determined at any point of the patch and in particular at the nodal points.

        4.2 Mesh data transfer

        A simplified data transfer procedure is obtained by nodal interpolation. The initial solution at the current mesh is interpolated from the solution of the previous mesh at the last time-step.

        5 Adaptive algorithms for FSI analysis

        Eq. (7) could be rewritten as:

        The second derivatives could be depicted in a tensor type notation as:

        Principal values and directions of the second derivative tensor can be calculated from:

        With the following two solutions,λ1(the minimum) andλ2(the maximum) values:

        whereX1andX2are the directions of the minimum and maximum principal values. For a uniform distribution of the interpolation error:

        The user sets the limits of minimum and maximum element sizes asdmin anddmax . Therefore, actual values ofhm¢in andh' maxat each node can be represented as:

        Fig. 2 illustrates the FSI staggered solution. In each several time steps, the fluid domain solution results are transformed into the adaptive algorithm process as of Fig. 3. After error estimation and re-meshing, the new mesh is applied to the FSI algorithm for the next time step.

        Fig. 2 Staggered FSI algorithm

        Fig. 3 Curvature based adaptive mesh generation algorithm

        Error factors.

        To evaluate the performance of transient-response histories with respect to a benchmark solution, a form of comprehensive error factor (C-error) given by Sprague and Geers (2006) is adopted:

        in which

        where

        In these equations,c(t) is a candidate solution in the form of a response history,b(t) is the corresponding benchmark history, andt1≤t≤t2is the time span of interest.Vis the magnitude error factor, which is insensitive to phase discrepancies, andPis the phase error factor, which is insensitive to magnitude discrepancies.

        Sprague and Geers (2006) setC>lt;0.1, 0.1≤C≤0.2 andC>gt;0.2 as the bounds for satisfactory, marginal, and unacceptable error, respectively. These bounds can be used to decide△Tadaptivethe maximum time period for mesh modification in specific problem where a reference solution is available otherwise a general empirical equation must be defined for determination of △Tadaptive. In this study a fine mesh is used as the reference solution. If the C-error at the end of total time exceeds the maximum acceptable value, a reduction in△Tadaptiveis needed. The following guideline is proposed for the maximum time period for mesh modification, △Tadaptivein a dynamic adaptive analysis.

        where

        In our experience for the maximum element size, six linear elements are considered to span the minimum wave length.According to Eq. (51), △Tadaptiveis the time that is required by wave front to span maximum mesh sizeLmas.

        6 Results and discussion

        Numerical examples of scatter wave propagation by adaptive finite elements are presented. Accuracy of the results using fine, equivalent and adaptive meshes is studied.The number of degrees of freedom (DOFs) in an equivalent mesh is approximately equal to the average number of DOFs in the adaptive meshes during the total analysis time. The latter procedure has been employed to assess the efficiency of the adaptive mesh.

        6.1 Concrete wall under a triangular pulse

        A concrete wall in contact with a semi-infinite reservoir is affected by a pulse of triangular plane wave. The results of adaptive and equivalent meshes are compared against that of a uniform fine mesh (as the reference solution). Fig. 4 shows the concrete wall with 0.3 m thickness and 2 m height.

        Concrete materialpropertiesconsist ofmassdensity of 2400 kg/m3,modulusofelasticity of21.0MPaandPoisson’s ratio of 0.2. The incident pulse has amplitude of 1.0 MPa and a duration of 1.0 millisecond as defined in Fig. 5.

        Fig. 4 Concrete wall under a triangular pulse

        Fig. 5 Pressure loading at the stand-off point

        Fig. 6 Power spectral density of triangular loading

        An incident plane pulse wave is considered inside the reservoir to represent a wave travelling toward the standoff point in the middle height of the wall. A non-reflective boundary condition is considered at just 2 m away from the wall in order to model the infinite boundary. The power spectral density of pressure incident wave has been plotted in Fig. 6 and the frequency content is determined.

        Accordingly, the maximum element size is 12 cm for the wave front but it could be increased elsewhere.

        Generally, by medium frequencies (MF) one could refer to the range of 300 kHz to 3 000 kHz. Frequencies below and higher than this range could be denoted as low (LF), and high(HF) frequencies, respectively. According to Fig. 6, the present problem can be categorized as an LF problem. The initial mesh shown in Fig. 7(a)–(c) and Fig. 7(d) is fairly fine.In Table 1, the number of nodes and elements in adaptive steps has been shown.

        Fig. 7 Successive updated meshes

        Fig. 8 Fine mesh with 1 877 nodes

        Fig. 9 Equivalent mesh with 609 nodes

        For the initial and reference uniform fine meshes as shown in Fig. 8, the maximum size of the elements is considered as 10 cm. In this case the fluid and structure interface nodes are kept coincident. The equivalent mesh is shown in Fig. 9. The total time required the scattered wave to travel from the wall to the infinite boundary is about 1.4 ms, after this time, the wave front exits the fluid domain and the pressure magnitude is decayed. According to Eq. (51), this period is divided into four 0.5 ms time steps and adaptation is performed once in each division

        In Figs. 10 to 12, adaptive steps are shown for the fluid domain.

        When the wave front is near the wall (Fig. 10), pressure is an element is justified. A coarse mesh has been generated near the infinite boundary atx=2, while a fine mesh is necessary around the wave front, as shown in Fig. 11.

        As the wave front is transmitted towards the infinite boundary, refinement is concentrated near this boundary. In Fig. 12 in the region near the wall a coarse mesh can be sufficient that has a non-essential limitation of at least one fluid element being attached to each structural element. Fig.13 depicts the time history of the scattered pressure for a middle point of reservoir. The existing jumps in the “adaptive”curve are due to the particular interpolation scheme for the solution data transfer from the old to the new mesh. Table 2 compares the global C-error factor and the CPU time evaluated for different meshes. The pressure error norm is improved at about 18 percent compared to that of the equivalent mesh. The adaptive mesh CPU time also includes the mesh updating time. This result showed improvements in both precision and CPU time when using the adaptive mesh algorithm. Figs. 14 and 15 clearly showed that the overall response of the wall is improved when using the moving adaptive mesh rather than the equivalent uniform mesh.

        Table 1 The number of nodes and elements in different meshes

        Table 2 Comparison of error factors and CPU times

        Fig. 10 Pressure contour at 0.5 ms

        Fig. 11 Pressure contour at 1 ms

        Fig. 12 Pressure contour at 1.5 ms

        Fig. 13 Scattered pressure time history in the middle of reservoir

        Fig. 14 Time history of acceleration in the x direction at the standoff point on the wall

        Fig. 15 Time history of velocity in the x direction at thestandoff point on the wall

        6.2 Concrete shell under step pulse

        An infinite 0.15 m thickness concrete shell, in contact with a semi-infinite reservoir, is affected by a pulse of step plane wave, as depicted in Fig. 16.

        The results of adaptive and equivalent meshes are compared with that of a uniform fine mesh. The incident pulse has amplitude of 1 MPa and a 1 μs rise time as shown in Fig. 17. The source of the pulse is at the center of the reservoir and the location of standoff is at the middle of the concrete shell. The power spectral density of pressure incident wave has been plotted in Fig. 18. The maximum element size is calculated as:

        With this result it can be derived that a very fine mesh is required. However, for the sake of comparison of adaptive and equivalent meshes, some level of error withfmax=2 000,which is related to 0.95 of maximum power spectral density is accepted.Lmax=10 cm and the initial mesh is generated according to Fig. 19.

        Table 3 presents the mesh data in different adaptive analysis steps. The guideline in Eq. (51) proposes 0.41 s for the mesh updating time, withLmax=10 cm, but for higher accuracy △Tadaptiveis set 0.2 s. Figs. 19 and 20 show the initial and the equivalent meshes, respectively.

        Table 3 Number of nodes and elements in different meshes

        Fig. 16 Plane strain concrete shell under step pulse

        Fig. 17 Pressure loading at the standoff point (rise time=1 μs)

        Fig. 18 Power spectral density of step loading

        Fig. 19 Initial mesh with 1 452 nodes

        Fig. 20 Equivalent mesh with 399 nodes

        Fig. 21 shows successive updated meshes based on the curvature indicator criterion. When the scattered wave front propagates almost in radial direction, the fine part of the mesh moves with it until it reaches to the infinite boundary. SinceLmaxis very small and a relatively large element size is still used for the fine mesh, oscillations are expectedly observed even in the fine mesh. This is illustrated in Fig. 22. The effect of these oscillations is not considered in the present comparison of adaptive and equivalent meshes.

        Fig. 21 Successive updated meshes and pressure

        According to Table 4 the pressure C-error factor of adaptive mesh is 11 percent less than that of the equivalent mesh. Also, CPU time has been reduced when using adaptive mesh. By comparing Table 2 and Table 4 it can be seen that the adaptive mesh is more effective for higher frequency contents of the incident wave loading.

        The time histories of acceleration and velocity are plotted in Figs. 23 and 24.

        Fig. 22 Time history of pressure at the standoff point

        Fig. 23 Time history of acceleration in x direction at the standoff point on the wall

        Fig. 24 Time history of velocity in x direction at the standoff point on the wall

        Adaptive and equivalent C-error factors are also compared with that of the fine mesh in Table 5. This showed that the velocity error factor of the adaptive mesh is 12.5 percent less than that of the equivalent mesh. This is an indication of the efficiency of dynamic adaptive solution.

        Table 4 Comparison of error factors and CPU times

        Table 5 Comparison of horizontal velocity error norms

        7 Conclusions

        This paper has presented a dynamic adaptive finite element analysis strategy for acoustic wave propagation due to underwater explosion in fluid-structure interaction problems.This approach is based on the C-error factor, which has an appropriate adaptive mesh refinement that is carried out at certain time intervals during the total time of wave propagation. It has been numerically illustrated that the error indicator based on the curvature could produce improvement as high as 22 percent in C-error factor for the pressure time history for the test cases considered. Similar results have been obtained for the response of velocity and acceleration.The curvature error indicator can be an effective tool for examining the adequacy of a finite element meshes by identifying the regions where mesh refinement is necessary in acoustic adaptive dynamic analysis. The proposed guideline for the mesh updating time interval can be used in the dynamic adaptively process and its performance may be investigated in future studies. The same procedure can be used with gradient based formulation for very low frequency waves. Therefore, reduction in computational cost and increase in accuracy for high frequency wave is the obvious improvement compared to other traditional works.

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