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  2. Perpustakaan Jeneral Tun Ibrahim
  3. Adaptive Mesh Refinement In Time Domain Numerical Electromagnetics Sarris Costas D
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Skickas inom vardagar. This book presents a method that allows the use of multiresolution principles in a time domain electromagnetic modeling technique that is applicable to general structures. The multiresolution time-domain MRTD technique, as it is often called, is presented for general basis functions. Add co-authors Co-authors. Upload PDF. Follow this author. New articles by this author. New citations to this author. New articles related to this author's research. Email address for updates. My profile My library Metrics Alerts. Sign in. Get my own profile Cited by View all All Since Citations h-index 21 16 iindex 39 In conclusion, one can consider the proposed method as an effective alternative for MRTD mesh termination, since it employs existing UPML implementations, linking them to MRTD codes via the simple connection algorithm that was earlier described.

The reason for that is that ABCs impose mathematical conditions at planes that include only a fraction of the equivalent grid points contained within a terminal cell. The concept of an interface-based solution to this question is depicted in Fig. Because of this trade-off, this mesh truncation method is more suitable to low-order MRTD schemes. Results from a one-dimensional implementation of these concepts are shown in Figs. Therefore, the thickness of the FDTD region for each case study is: 0.

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That is also the reason for the phase difference of the reflected waveforms in Fig. Finally, Fig. Obviously, the obtained ABC performance is similar in all MRTD cases and confirms the capability of the method to provide an efficient means of implementing an ABC-truncation of wavelet grids. Hence, the developed algorithm constitutes a computationally efficient tool for jointly exploiting the advantages of FDTD and MRTD, which is critical for the acceleration of time-domain schemes, when applied to large-scale problems of current microwave technology structures.

The previous expressions are readily programmable and allow for the development of arbitrary-order Haar wavelet MRTD codes. Yet, code efficiency is greatly enhanced by a priori recognizing the non-zero coefficients D0 , D1 , D2 , D3 and omitting operations that involve multiplications by zero in the main time-stepping loop. This is done at the pre-processing stage of an MRTD code.

The main attractive feature of wavelet-based, time-domain techniques is the simple implemen- tation of adaptive meshing, through the application of a thresholding procedure to eliminate wavelet coefficients that attain relatively insignificant values, at a limited compromise of ac- curacy. However, little attention has been devoted so far to the investigation of computational costs and accuracy trade-offs in order to obtain thresholding-related operation savings. This chapter presents an efficient implementation of thresholding applied to a nonlinear problem and reports significant execution time savings compared to the conventional FDTD technique, that the application of the proposed method has led to.

A motivating force for this research activity is the fact that wavelet-based methods provide the most natural framework for the implementation of adaptive grids, dynam- ically following local variations and singularities of solutions to partial differential equations. In particular, it can be proven that the decay of wavelet expansion coefficients of a square integrable function depends on the local smoothness of the latter [32]. Hence, significant wavelet values are expected at space—time regions, where high variations in the numerical solution evolve.

In this sense, sparing the arithmetic operations on wavelet coefficients below a certain threshold—small enough according to accuracy requirements—amounts to imposing coarse gridding conditions at those regions, while allowing for a denser mesh at parts of the domain where the solution varies less smoothly. Several approaches to wavelet-based mesh refinement have been presented in the litera- ture. However, the resulting method was strictly equivalent to a subgridded FDTD and presented obvious accuracy disadvantages, since it was not enhanced by the interpolatory operations, typi- cally employed in subgridded FDTD [44].

In general, the necessity to incorporate wavelets into a time-domain simulation technique is always related to dynamic rather than static subgridding, since the latter—having been extensively studied in the literature—can be nowadays efficiently implemented by several existing engines.

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On the contrary, adaptive, wavelet-based meshing was introduced in [32] and applied to electromagnetic structures in [12—15], always in conjunction with high-order basis functions, these being either Daubechies or B-splines. The current work meets this challenge by building up a two-level scheme on the simplest wavelet basis, the Haar basis and attempting a simple and explicit implementation of an algorithm for the thresh- olding of wavelet coefficients, based on ideas originally related to shock-wave problems of computational fluid dynamics.

Execution time measurements for the algorithm as applied to a nonlinear optics problem, show that this procedure can actually lead to faster-than-FDTD simulations. The resultant plots, for wavelets of orders 0—3 are shown in Figs. Evidently, despite small differences between them, all four curves have a similar pattern: Around time step , unthresholded wavelet coefficients are doubled, as a result of the pulse incidence on the dielectric slab, that generates an additional reflected wavefront.

Finally, the absorption of the reflected wavefront signals the end of the simulation and the decay of the number of unthresholded coefficients to almost zero. Overall, thresholding of wavelet coefficients yields a compression in memory requirements by Also, Fig. Third-order wavelets.

S11 [dBs] S11 [dBs]. S11 for the dielectric slab, as computed via adaptive MRTD. The excellent comparison of the numerical results to transmission line theory, demonstrates that wavelet thresholding has a limited effect on the accuracy of the algorithm. Yet, this aspect of wavelet adaptivity will not be further investigated in this work, because taking advantage of wavelet compression at runtime, presupposes the application of memory operations re-shuffle of field arrays that have an adverse effect on execution time.

It is noted though, that there are applications, where field values at all time steps for a certain domain need to be stored in memory. A posteriori processing those and wavelet thresholding them, can lead to significant memory economy [45]. Moreover, Figs. The source is located in the middle of the domain and produces two symmetric, left- and right-propagating waves.

Two observations are now in order: First, wavelets follow successfully the field wavefronts, throughout the domain. Second, in order to keep track of the evolution of wavelet coefficients, one has to essentially study the movement of the boundaries of the unthresholded wavelet regions. This task is limited to tracking the direction of the velocity of those boundaries. In the following, these observations are exploited for the efficient application of thresholding in MRTD.

Front boundaries 1. Scaling cell Scaling cell 1. Front boundaries. This nonlinear phenomenon is based on a self-phase modulation SPM induced negative dispersion that the pulse experiences along the fiber and was originally reported and studied in [46]. Scaling cell Scaling cell. Numerical modeling of this case was also pursued in [47], by means of the Battle—Lemarie cubic spline-based S-MRTD technique, resulting in execution time that was reported to be larger than FDTD by a factor of 1.

In this work, a one wavelet-level Haar MRTD scheme is used, for the purpose of estimating the improvement in computational performance that exclusively originates from the dynamic adaptivity of MRTD rather than the high-order of an underlying scaling basis. The FDTD update equations for the system of 3. For these functions, definitions 2. Upon substitution of 3. It is worth mentioning that operations in 3.

Then, scaling and wavelet terms of the forward field are updated at the excitation cell, as:. Similarly, the hard boundary condition on the backward wave at the nth cell of the domain Fig. Then, the scaling and wavelet terms of the backward field, at the nth MRTD scaling cell, are updated according to the scheme:.

The absorbers are implemented as in [47], by expanding their quadratically varying conductivities in Haar scaling functions. Simu- lation data for both aforementioned cases are provided in Table 3. The two methods agree well on the peak of the transmitted intensity which cor- responds to the forward wave intensity at the end of the fiber. The slight difference can be attributed to the fact that in the MRTD absorber, the conductivity was assumed constant within each scaling cell and therefore it varied less smoothly than the corresponding FDTD.

Moreover, Fig. A similar pattern for wavelet behavior was obtained in [15], where nonuniform multiconductor transmission line equations were solved via a biorthogonal wavelet basis. In the following, this observation is utilized for the devel- opment of a computationally efficient approach to the problem of thresholding of wavelet coefficients.

For this purpose, the following method is adopted [12, 15, 32], for both forward and backward propagating wave arrays:. In all operations, only scaling and active wavelet terms are taken into account. The corresponding cells remain active only if their wavelets are above the threshold. Since only one wavelet level is used here, this implies cells that are immediately to the left or to the right of cells on the border of active regions. Hence, thresholding checks and update operations are limited to a subset of the field coefficients. Note that the use of the Haar basis and a single wavelet level scheme, keeps the implementation of this adaptive algorithm relatively simple and readily expandable to three dimensions.

On the other hand, the complexity of numerical solvers based on higher-order basis functions and mul- tiple wavelet levels has regularly undermined the potential of adaptivity to yield execution times better than FDTD. Furthermore, the frequency of thresholding checks is implicitly dependent on the CFL number s of the simulation, which effectively determines the maximum number of cells that a front may move through in a single time step. Physically, the thresholding algorithm, that was first introduced for the numerical solution of shock wave problems in [32], assumes the evolution of wavelet coefficients along wavefronts defined from the characteristics of a given problem.

Then, the purpose of adding pivot elements, to extend the domain of active coefficients, is actually the tracking of these wavefronts as they move throughout the computational domain. The concept of this algorithm is schematically explained in Fig. As shown, adding the pivot elements at the two edges of a wavefront, ensures the tracking of the direction of its movement. Then, the active wavelet region moves to the left, with the wavefront. Otherwise, it follows a movement of the front to the right. To this end, the second case that was presented in Section??

In all cases, thresholds up to 0. In Figs. The last two are in good agreement both with each other and with the previously presented FDTD and MRTD results, while the first suffers from significant numerical errors, that demonstrate themselves as a ripple corrupting the pattern of the waveform.

Therefore, the fact that an accelerated performance of adaptive MRTD with respect to FDTD was achieved is important and demonstrates the potential of the algorithm for larger geometries. The satisfactory performance of the proposed technique stems from its relative simplicity that allows for an efficient implementation of its two components: thresholding tests of wavelet coefficients and operation savings while performing updates of field arrays.

This combination allows the resulting technique to adapt to the problem at hand, opti- mally distributing computational resources in a given domain as needed, by recursively refining a coarse grid in regions of large gradient of electromagnetic field energy. Thus, instead of setting up a statically adaptive mesh, which is the subject of extensive previous research, the idea of a dynamically adaptive mesh is pursued, because of the optimal suitability of the latter to the na- ture of time-domain electromagnetic simulations.

However, the algorithm is still implemented within the context of FDTD, with second-order accurate update equations and is referred to as dynamically AMR-FDTD, to emphasize the dynamic evolution of the underlying mesh. This chapter is aimed at providing the reader with a detailed description of the algorithm, filled with all the information needed for its two- and three-dimensional implementation.

It is especially suitable for wide-band applications since it allows for the characterization of a given structure in a broad frequency range, through a single simulation. However, the FDTD stability and dispersion properties impose severe limitations on the choice of the cell size and the time step of the method, rendering its application to complex structures computationally expensive. The challenge of accelerating FDTD simulations for practical geometries has been ad- dressed in the past with a variety of static subgridding techniques [10, 44].

For example, the presence of metallic edges, or high dielectric permittivity inclusions, would call for a locally dense mesh, embedded in a coarser global one. The use of local mesh refinement typically results in significant computational savings com- pared to the conventional FDTD, despite the fact that its implementation is associated with additional interpolation and extrapolation operations in both space and time.

However, this approach ignores the dynamic nature of time-domain field simulations. In fact, techniques such as FDTD and TLM essentially register the history of a broadband pulse propagating in a device under test, along with its multiple reflections from parts of the latter. Hence, a sharp edge of a microstrip structure is not continuously illuminated by the pulse excitation; on the contrary, it is so for a potentially small fraction of the total simulation time, during which a local mesh refinement around it is needed.

Therefore, static mesh refinement, which is widely employed in frequency-domain simulations and has been incorporated in com- mercial finite-element tools, is only a suboptimal solution to the mesh refinement problem in the framework of time-domain analysis. More recently, a moving-window FDTD MW-FDTD method was proposed for the tracking of the forward propagating wave in the two-dimensional terrain environment of a wireless channel [49], similar to ideas previously proposed in [50, 51]. The single moving window used by the method was characterized by fixed size and velocity and therefore, it could not track reflections which were absorbed by terminating boundaries of the window.

As a result, the MW-FDTD is not well-suited for microwave circuit simulations, where the modeling of phenomena as common as signal reflection and branching would require multiple and potentially rotating windows. In the context of computational fluid dynamics, the technique of adaptive mesh refinement AMR was introduced in [52], for the solution of hyperbolic partial differential equations.

The application of AMR is based on the use of a hierarchical mesh, recursively developed through the refinement of a coarse root mesh, which covers the entire computational domain. The regions of the computational domain that need further mesh refinement are detected via error estimates or indicators such as gradients of the quantity to be solved for. There may also be dense mesh regions, where the use of a dense mesh is not necessary after a certain time step.

These can then be coarsened, again in a recursive manner.

Adaptive Mesh Refinement In Time Domain Numerical Electromagnetics Sarris Costas D

Dense and coarse mesh regions are organized via a clustering algorithm that is accompanied by regular checks every certain time steps of the error estimates, which guide the process of migration of a cell from one level of resolution to another. This procedure can be associated with the algorithm of the previous chapter, which used wavelet field expansions in order to track the spatiotemporal evolution of shock-wave and nonlinear optical pulse propagation problems, respectively.

Instead of using fixed subgrids, this method uses subgrids that are adaptively defined, according to the evolution of field distributions in space and time. As an example, when a Gaussian pulse propagating along a microstrip line is simulated, the adaptive mesh refinement scheme successfully tracks the movement of the pulse, thereby refining only the region that surrounds the propagating pulse. Section 4. Finally, Section 4. Throughout the algorithmic development of the AMR-FDTD, it will be ensured that these subregions can share planar boundaries, yet they cannot overlap.

This is important in order to preserve the possibility of further refinement of these subregions independently from each other, as required by the evolution of the field solu- tion. Hence, a refinement factor of 2 is used in every direction, reducing the Yee cell volume of the initial mesh by a factor of 8. To summarize, a coarse mesh has been defined in the rectangular region A, enclosing finer meshes in rectangular subregions Bn of the latter. The mesh of region A, henceforth referred to as mesh A, will be called the root mesh, or level-1 mesh.

The meshes of regions Bm , or meshes Bm , will be called child meshes of A, or level-2 meshes. Recursively, each Bn can be further refined to have its own child meshes, again refining the cell sizes involved by a factor of 2. An example is shown in Fig. There is only one level-1 mesh, which is also called the root mesh and covers the entire computational domain. Thus, the two meshes form a child—parent relation. The child meshes of the same parent may share an edge, but they may not overlap otherwise.

Hz Ey Hx Ex Ez. The sampling points of a parent mesh may coincide or not with the sampling points of its child mesh, depending on whether the refinement factor is odd or even, respectively. In this work, the refinement factor is 2 or powers of 2, with respect to the root mesh , hence the grid points of child and parent meshes do not coincide.

An alternative case, where the choice of a refinement factor of three renders the parent mesh sampling points also child mesh sampling points can be found in [44]. Furthermore, the Courant number is fixed to a constant value s in all meshes. Applying 4. For example, level-2 meshes are updated twice as many times as the root mesh. Thus, another shortcoming of the conventional FDTD is addressed; the minimum time step of the algorithm is only used for the update of regions of large field variations, as opposed to the whole domain, a salient feature that is also part of the fixed subgridding algorithms of [10, 44].

Check the number of time steps executed. If it is an integer multiple of NAMR , perform adaptive mesh refinement to create a new mesh tree, and carry the field values from the old mesh tree to the new mesh tree. Update fields of the root mesh. Copy fields from the root mesh to the boundary of the child meshes. Copy fields from child meshes back to the root mesh, for the time steps of the latter. Check whether the maximum time step has been reached. If so, terminate the simula- tion, otherwise return to step 1.

In this section, the types of interfaces that occur in an AMR-FDTD domain are presented, along with their treatment in the update process.


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To facilitate the presentation of the different cases, a two-dimensional case is discussed readily extensible to three dimensions. Consider two child meshes B1 and B2 , embedded in a root mesh A. Three separate cases of boundaries can be identified:. Segments ab, bc, de: These are boundaries between child and parent meshes CPB. Evidently, it is possible that a boundary may belong to more than one of the aforementioned categories. Then, its classification is based on the following hierarchy:. The steps outlined here can be recursively extended to mesh trees of more levels, by considering, for example, level-2 meshes as roots for level 3-meshes and so on.

Then, the following procedure is applied. Obviously, these updates are nontrivial, since they invoke grid points of the root mesh, calculated at time steps of the child mesh. Therefore, interpolation needs to be carried out, in a way that is analyzed in the next subsection.

This process can be readily extended to the multilevel case, where meshes of levels 1, 2, 3,. It should be noted that all meshes of the same level need to be updated for a certain time step before advancing to the next time step, because their values are invoked in the update equations of neighboring meshes. The following operations are recursively performed:.

For each mesh of level L, the magnetic field components are updated. For each mesh of level L, the electric field components are updated. The aforementioned steps can be implemented in a function upd ate Le vel L, S , which is recursively called to update all meshes with level greater or equal to L and a local time step S.

The main program just needs to call upd ate Le vel 1, 0 for the update of the entire mesh tree for one time step. Since such a boundary is characterized by fixing one spatial variable and letting the other two vary, along with time, trilinear interpolation provides the expression employed in all these updates.

Since the child mesh regions Bn belong to their root mesh region A, the use of 4. Therefore, the treatment of SB-type bound- aries can be limited to boundaries between child meshes of the same level. Since, by Yee cell convention, the electric field is sampled at cell edge centers and the magnetic field is sampled at cell face centers, two meshes of the same level share tangential electric field and normal magnetic field components at the interface between them see, e.

Adaptive Mesh Refinement (AMR) through a stationary valve

Referring to Fig. In particular, the index of Hy1 in mesh 2 should be transparent to mesh 1.

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For this purpose, the positions of the SB-type boundaries between child meshes of the same level are recorded in a table, after each mesh refinement i. Hy2 mesh1 y Ex Hz1 Hz2 mesh2 Hy1. Ideally, an SB-type boundary should be transparent and meshes linked by SB-type boundaries connecting meshes of the same resolution should. Figure 4. A straightforward solution to the problem of handling these junctions, that in dielectrically homogeneous domains operates acceptably well, is to treat them as CPB-type boundaries.

This approach is employed in the microwave circuit examples of the Chapter 5. However, for the analysis of structures such as the inhomogeneous waveguides of Chapter 6, the following approach is proposed. As shown in the figures, the update of an electric field node E-node requires four neighboring magnetic field nodes H-nodes , some of which may not belong to the same mesh.

Let us call those external H-nodes. It should be noted that an E-node on a boundary may be shared by 2, 3, or 4 neighboring meshes of the same level. It only needs to be updated in one mesh and then copied to all the other meshes. To avoid double updates, the following three situations are considered:. If an E-node on an SB-type boundary is not at a corner of at least one neighboring mesh, it should be updated in that mesh by using one external H-node from the other mesh, then copied to all other meshes. For example, in Fig. If an E-node on an SB-type boundary is at a corner of all the neighboring meshes, the situation can be further divided in two subcases: a There is at least one pair of meshes forming a cross-junction around the E-node.

For example, since meshes A and C form a cross-junction at the E-node c , c can be updated in A by using two external H-nodes from C, or vice versa. Then, c is copied to all the un-updated neighboring meshes. None of the neighboring meshes can obtain two external H-nodes, therefore this E-node has to be treated as a CPB-type boundary and updated accordingly. These conditions are enforced in both root and child meshes.

However, any other type of boundary condition can be readily incorporated. Perfectly matched layer PML -type conditions, would simply extend the com- putational domain by the number of the absorber cells. Field samples stored in the old mesh tree are transferred to the new mesh tree. This is straightforward for the root mesh, since the new tree has the same root mesh as the old tree. On the other hand, the possibility of an overlap between any child mesh of the new tree and any child mesh of the old tree is checked. If there is such an overlap, the field samples in the overlapping region are transferred from the old to the new child mesh.

For example, consider the situation shown in Fig. These are maintained and kept available, according to the proposed algorithm of Section 4.

Example 1: Laplace Problem

Again, trilinear interpolating operations are employed to initialize the new mesh. It is finally noted that source conditions are always enforced in the root mesh. If a child mesh overlaps with the source region, the overlapping part of the source should also be enforced in the child mesh. Standard interpolation operations, which are the common characteristic of any subgridding algorithm have been proposed. What distinguishes AMR-FDTD from previous subgridding approaches is the adaptive mesh refinement, which enables the adaptive movement of the subgrids.

Then, the gradient of the energy is numerically approximated by a second-order finite-difference expression, as:. Therefore, the adaptive mesh refinement is executed in a cell when both an instantaneous and a calculated over the whole simulated time threshold are exceeded. On the other hand, as the AMR-FDTD simulation begins, all the electric field compo- nents assume zero values, except for the ones excited by the source.

The detection of cells to be refined at this time-marching stage is difficult, since the energy gradient is too small to surpass the threshold set. To overcome this difficulty, in addition to the cells detected by thresholding, the cells in the source region are also marked for refinement, for a certain period of time. For the multilevel case, the same algorithm is recursively applied at each mesh level. Assuming again that NL is the maximum number of levels, the mesh regeneration and, subse- quently, the update of the mesh tree is pursued as follows:.

If no cells are marked, we return to the previous step. For L varying from 2 to the number of levels of the new mesh tree, each level-L mesh is initialized as follows: if it overlaps with any level-L mesh of the old mesh tree, the electric and magnetic field components are obtained from the overlapping region. For the remainder of the mesh which does not overlap with any other previous child mesh the electric and magnetic field components are calculated by interpolating their values from the corresponding parent mesh. The previous mesh tree is then removed from memory. The only mesh that is not subject to regeneration every NAMR time steps is the root mesh.

However, this refinement process takes place every NAMR time steps to avoid loading every time step with the mesh refinement operations. Note that no assumption is being made as to the direction of the wave velocity, which in general is unknown. The effect of the spreading factor, as well as the thresholds defined in the previous section will become evident in the numerical results of the next two chapters.

This procedure is called clustering. To evaluate the quality of clustering, the box coverage efficiency is introduced, defined as the ratio of the total volume of the marked cells, covered by a box, to the volume of the box. For the implementation of this clustering procedure, the methodology proposed in [55] is followed. At the beginning, the bounding box enclosing all the marked cells is found and its coverage efficiency is calculated. Each of the new boxes is shrunk to just cover the marked cells.

This iterative process continues until the maximum number of boxes is reached. The black dots represent the marked cells. Box A is the root mesh. The determination of the position of the cut plane is detailed in [55]. Therefore, the comments on the accuracy of the technique are meant to be always referred to such an FDTD scheme.

Let us also define the AMR-coverage as the ratio of mesh-refined regions to the total volume of the computational domain. In general, decreasing the AMR-coverage will reduce the execution time of the code but will also reduce its accuracy, since the mesh becomes coarser overall. In- creasing this parameter leads to less AMR-related operations. A subtle side-effect of a large NAMR is the following: newly generated mesh-refined regions are always extended by a fac- tor proportional to NAMR see 4.

Hence, the coverage increases and thus the execution time. In general, the recommended values of NAMR are between 10 and The most important source of errors in any static or dynamic mesh refinement scheme is the reflections at a CPB-type boundary. Note that a large value of the latter enforces the generation of more smaller meshes, improving the AMR-coverage.


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  • At the same time, it generates more CPB- and SB-type boundaries, thus increasing the operations related to their management. These guidelines and the inherent trade-offs in the choice of the parameters are further illustrated in the numerical results of the next two chapters. However, it does implement a multiresolution, space and time adaptive moving mesh in three dimensions, re- generated every certain time steps.

    While the adaptive MRTD operates on nonthresholded field wavelet coefficients, that may be spatially and irregularly distributed, the AMR-FDTD clusters the Yee cells that need mesh refinement and encloses them in rectangular subgrids. This leads to a much more systematic mesh refinement process. In the MRTD implementations that are included in this monograph, as well as in the vast majority of the MRTD literature, a uniform time step is set for the update of all field scaling and wavelet coefficients, regardless of their resolution.

    An important step toward this end is the work reported in [56]. In fact, the realistic, two- and three-dimensional applications presented in the next two chapters are aimed at highlighting the former two qualities of the AMR-FDTD. On the other hand, MRTD is a class of numerical techniques that can achieve with the use of higher-order basis functions, as opposed to the Haar basis high-order of accuracy, paying the price of complicated enforcement of source and boundary conditions. Hence, despite its more recent introduction the dynamically AMR-FDTD technique is closer to being considered as a mature technique compared to MRTD, which is still related to important and challenging questions when it comes to practical applications.

    The dynamically adaptive mesh refinement FDTD technique of the previous chapter is now applied to realistic microwave circuit geometries, namely a microstrip filter, a branch coupler, and a spiral inductor. Through these applications, the salient properties of the technique along with its excellent potential to dramatically reduce FDTD simulation times are shown. Trade- offs between speed and accuracy involved with the application of the algorithm to problems of interest are also discussed. First, what is the effect of the various parameters, defined in the previous chapter on the performance and accuracy of the algorithm and second, whether the overhead that AMR-FDTD accumulates from the application of the mesh refinement can still leave some room for computational savings stemming from a reduced overall number of opera- tions.

    Both questions are negotiated through the application of the technique to three microwave circuit geometries, namely, a microstrip low-pass filter, a branch coupler, and a spiral inductor. In all cases, a maximum of two mesh levels is implemented or a maximum mesh refinement factor of 8 for a Yee cell of the root mesh. In all these experiments, a two-level AMR-FDTD is implemented, in order to facilitate the presentation of the effect of its parameters on accuracy and execution time.

    All simulations were executed on an Intel Xeon 3. A voltage source excitation is imposed at 3 mm from the edges. In all simulations, a Courant number of 0. To demonstrate the evolution of the child meshes over time, the ratio of the volume of mesh-refined areas occupied by child meshes to the total volume, referred to as AMR- coverage of the domain and the number of child meshes as a function of time steps are shown in Fig. In addition, Figs.

    As the fields impinge upon the absorbing boundaries of the structure, the field values in the working volume of the domain decrease. Consequently, the spatial field variation be- comes smoother, which causes the AMR-coverage to decrease. Figure 5.