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Fig. 11. (a) Schematic two-dimensional (2-D) representation (not scaled) of the 3-D computational domain with extremely unstructured mesh used for the simulation of plane-wave propagation. (b) Free-space 3-GHz plane wave simulated in the unstructured mesh with local time steps.

Figure 11 (a) Schematic two-dimensional (2-D) representation (not scaled) of the 3-D computational domain with extremely unstructured mesh used for the simulation of plane-wave propagation. (b) Free-space 3-GHz plane wave simulated in the unstructured mesh with local time steps.

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where the overbar indicates an average over one time step. These two equations permit to write “new” volume-averaged field components (LHS) as a function of “older” volume-aver- aged values (first term on the RHS) and surface-averaged field components averaged over one time step (second term on the LHS). Time Stepping: The fundamental FVTD equations (2) or (4) are discretized in space. The field components in theses equa- tions are time-dependent vector functions [10]. The FVTD up- date equations are obtained introducing time steps denoted by an index n and separated by a period At. After integrating (4) during one full time step, we can write
where the overbar indicates an average over one time step. These two equations permit to write “new” volume-averaged field components (LHS) as a function of “older” volume-aver- aged values (first term on the RHS) and surface-averaged field components averaged over one time step (second term on the LHS). Time Stepping: The fundamental FVTD equations (2) or (4) are discretized in space. The field components in theses equa- tions are time-dependent vector functions [10]. The FVTD up- date equations are obtained introducing time steps denoted by an index n and separated by a period At. After integrating (4) during one full time step, we can write
Fig. |. Two-dimensional illustration of the two stages of one FVTD iteration step. (a) Barycenter values to face values. (b) Face values to barycenter values.
Fig. |. Two-dimensional illustration of the two stages of one FVTD iteration step. (a) Barycenter values to face values. (b) Face values to barycenter values.
Fig. 2. Illustration of the two steps of the Lax—Wendroff time-stepping scheme for numerical solution of a differential equation of the type 0u/8t = f(w). The gray dots indicate when the function f(w) is evaluated.
Fig. 2. Illustration of the two steps of the Lax—Wendroff time-stepping scheme for numerical solution of a differential equation of the type 0u/8t = f(w). The gray dots indicate when the function f(w) is evaluated.
Fig. 3. Two-dimensional representation of the subdomain partition of a computational domain containing two small structures (not shown). The transition from regions with time step At to regions with time step 4At requires an intermediate zone with time step 2At. The boundaries (including adjacent cells) between contiguous domains are strictly separated so that they can be described as two-level systems. determined through the maximum of criterion (6) on the entire mesh as follows:
Fig. 3. Two-dimensional representation of the subdomain partition of a computational domain containing two small structures (not shown). The transition from regions with time step At to regions with time step 4At requires an intermediate zone with time step 2At. The boundaries (including adjacent cells) between contiguous domains are strictly separated so that they can be described as two-level systems. determined through the maximum of criterion (6) on the entire mesh as follows:
Fig. 4. Two-dimensional illustration of the neighborhood of a considered triangle C’ (tetrahedron in 3-D) defined by its nodes (black dots). The direct neighborhood is formed by the cells V1 that share a face with C’. The extended neighborhood contains in addition the cells N2 that share at least one node with C. Fig.5. Two-dimensional representation of the computational domain partition according to the local time-step level. The small digits in the cells indicate the multiple of the fundamental time step At that could be used according to condition | only (stability condition for the single cell). The cells with white circles are not upgraded to level 3 because of condition 3.
Fig. 4. Two-dimensional illustration of the neighborhood of a considered triangle C’ (tetrahedron in 3-D) defined by its nodes (black dots). The direct neighborhood is formed by the cells V1 that share a face with C’. The extended neighborhood contains in addition the cells N2 that share at least one node with C. Fig.5. Two-dimensional representation of the computational domain partition according to the local time-step level. The small digits in the cells indicate the multiple of the fundamental time step At that could be used according to condition | only (stability condition for the single cell). The cells with white circles are not upgraded to level 3 because of condition 3.
Fig. 6. Update schedule for a computational domain partitioned with three time-step levels. The P’s indicate predictor steps and the C’’s indicate corrector steps. Simultaneously satisfying the three conditions for the partition displaces the boundary toward larger cells, therefore providing some reserve for the stability criterion. Conditions 2) and 3) avoid the application of larger local time steps on isolated large tetrahedrons, which would not increase the efficiency since it involves more (computationally more expensive) “boundaries” than “bulk.”
Fig. 6. Update schedule for a computational domain partitioned with three time-step levels. The P’s indicate predictor steps and the C’’s indicate corrector steps. Simultaneously satisfying the three conditions for the partition displaces the boundary toward larger cells, therefore providing some reserve for the stability criterion. Conditions 2) and 3) avoid the application of larger local time steps on isolated large tetrahedrons, which would not increase the efficiency since it involves more (computationally more expensive) “boundaries” than “bulk.”
Fig. 7. Schematic representation of a two-level boundary system between subdomains with different time steps. An arbitrary field component is denoted by £ and the iteration step by 7. The Lax—Wendroff predictor steps are denoted with dark grey arrows and the corrector steps with light gray arrows. The artificial half steps are indicated using black arrows.
Fig. 7. Schematic representation of a two-level boundary system between subdomains with different time steps. An arbitrary field component is denoted by £ and the iteration step by 7. The Lax—Wendroff predictor steps are denoted with dark grey arrows and the corrector steps with light gray arrows. The artificial half steps are indicated using black arrows.
Fig. 9. Influence of the local time stepping on the field component £, at two points located on either side of the time-step boundary. The wave is traveling from cells with linear dimensions \/10 to smaller cells with dimensions \/20. EF, represents the time-dependent field component simulated with local time steps and AF, the deviation from the simulation performed without local time steps. Fig. 8. (a) Top schematic representation of a parallel-plate waveguide with time-step boundary. Level-1 cells have linear dimensions approximately two times smaller than level-2 cells. Simulation of the wave incident from both sides has been performed. (b) 3-D representation of the cube outer surface mesh. The top and bottom boundaries are perfect electric conductors (PECs) and the left and right boundaries are perfect magnetic conductors (PMCs). The plane wave is polarized in the z direction.
Fig. 9. Influence of the local time stepping on the field component £, at two points located on either side of the time-step boundary. The wave is traveling from cells with linear dimensions \/10 to smaller cells with dimensions \/20. EF, represents the time-dependent field component simulated with local time steps and AF, the deviation from the simulation performed without local time steps. Fig. 8. (a) Top schematic representation of a parallel-plate waveguide with time-step boundary. Level-1 cells have linear dimensions approximately two times smaller than level-2 cells. Simulation of the wave incident from both sides has been performed. (b) 3-D representation of the cube outer surface mesh. The top and bottom boundaries are perfect electric conductors (PECs) and the left and right boundaries are perfect magnetic conductors (PMCs). The plane wave is polarized in the z direction.
Fig. 10. Frequency dependence of the numerical error introduced by local time-stepping. Varying the frequency for a given mesh corresponds to changing the discretization: the upper horizontal scale gives the typical discretizations on both sides of the time-step transition.
Fig. 10. Frequency dependence of the numerical error introduced by local time-stepping. Varying the frequency for a given mesh corresponds to changing the discretization: the upper horizontal scale gives the typical discretizations on both sides of the time-step transition.
Fig. 12. Measured speed-up as a function of the highest time-step level used.
Fig. 12. Measured speed-up as a function of the highest time-step level used.
Fig. 13. Time-domain dependence of the plane-wave electric field at two different points. (a) At the barycenter of a large cell outside the spheres. (b) At the barycenter of a cell inside the tiny inner sphere. The field component FE’, simulated using local time steps and the difference AF, to the reference simulation (without application of local time steps) are represented in the graphs. Fig. 15. Simulated radiation pattern of the DRA at 5.3 GHz in the £-plane (containing the probe and the axis of the hemisphere). The lower graph shows the difference (in decibels) between the simulations performed with and without local time stepping.
Fig. 13. Time-domain dependence of the plane-wave electric field at two different points. (a) At the barycenter of a large cell outside the spheres. (b) At the barycenter of a cell inside the tiny inner sphere. The field component FE’, simulated using local time steps and the difference AF, to the reference simulation (without application of local time steps) are represented in the graphs. Fig. 15. Simulated radiation pattern of the DRA at 5.3 GHz in the £-plane (containing the probe and the axis of the hemisphere). The lower graph shows the difference (in decibels) between the simulations performed with and without local time stepping.
Fig. 14. Discretization of the hemispherical DRA. Only the triangulation of boundary surfaces is displayed. The hemisphere radius is r = 12.5 mm and the permittivity ¢,, = 9.5. Both the length of the probe and its displacement from the center of the hemisphere are equal to 6.5 mm.
Fig. 14. Discretization of the hemispherical DRA. Only the triangulation of boundary surfaces is displayed. The hemisphere radius is r = 12.5 mm and the permittivity ¢,, = 9.5. Both the length of the probe and its displacement from the center of the hemisphere are equal to 6.5 mm.
Fig. 16. Time-domain dependent magnitude of the electric field at the barycenter of two different cells of the computational domain. (a) The cell is located inside the coaxial feed. (b) The cell is outside of the DRA (free space). The graphs represent the magnitude of the E-field simulated using local time stepping and the difference AF with respect to the reference simulation without application of local time steps.
Fig. 16. Time-domain dependent magnitude of the electric field at the barycenter of two different cells of the computational domain. (a) The cell is located inside the coaxial feed. (b) The cell is outside of the DRA (free space). The graphs represent the magnitude of the E-field simulated using local time stepping and the difference AF with respect to the reference simulation without application of local time steps.
SUMMARY OF RESULTS OBTAINED USING THE PRESENTED TIME-STEPPING SCHEME barycenter of two cells: The cell in Fig. 16(a) is (level 1) located in the coaxial cable near the port o and the cell in Fig. 16(b) is a large cell (level 4) lo a small cell f the antenna cated in free space outside the DRA. The differences AF’ between the simu- lations performed with and without local time step ping are also plotted in the graphs. The small magnitude of these differences demonstrates the very low level of error introduced by local time stepping. This fact is also reflected in the return loss (511) of the antenna, where differences between both simulations (with and without local time steps) are under 0.015 dB over t quency range. he entire fre-
SUMMARY OF RESULTS OBTAINED USING THE PRESENTED TIME-STEPPING SCHEME barycenter of two cells: The cell in Fig. 16(a) is (level 1) located in the coaxial cable near the port o and the cell in Fig. 16(b) is a large cell (level 4) lo a small cell f the antenna cated in free space outside the DRA. The differences AF’ between the simu- lations performed with and without local time step ping are also plotted in the graphs. The small magnitude of these differences demonstrates the very low level of error introduced by local time stepping. This fact is also reflected in the return loss (511) of the antenna, where differences between both simulations (with and without local time steps) are under 0.015 dB over t quency range. he entire fre-
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Finite Volume MethodsComputational ElectromagneticsMesh generationNumerical AnalysisFine Structure ConstantElectrical and Electronic EngineeringMaxwell EquationsFinite Difference Methods
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