Apparatus and method for transforming a coordinate system to simulate an anisotropic medium
A method and apparatus for simulating a mesh element of an anisotropic medium are provided. A unitary transformation is applied to an initial coordinate system of the mesh element by a transformation module to produce a transformed reference coordinate system of the mesh element. Maxwell's equations for the mesh element are solved by an update generation module using computational methods to obtain an electric field tensor and an electric displacement field tensor within the mesh element. A unitary transformation to the electric field tensor and the electric displacement tensor are performed by a transformation module to calculate a corresponding electric field tensor and electric displacement tensor for the mesh element in the initial coordinate system.
1. A computer-implemented method for simulating an electromagnetic property of a medium, the medium being meshed to define a plurality of mesh elements in an initial reference frame, the method comprising, for each mesh element:
determining a unitary transformation at a transformation module having a processor configured for transforming an initial permittivity tensor in the initial reference frame into a transformed permittivity tensor in a transformed reference frame, the transformed permittivity tensor being diagonal;
using a standard finite-difference time-domain (FDTD) process for simulating the electromagnetic property of the medium;
while modifying the standard FDTD process by:
applying the unitary transformation at the transformation module to an initial electric displacement field tensor in the initial reference frame to generate a transformed electric displacement field tensor in the transformed reference frame;
communicating the transformed permittivity tensor, the transformed electric displacement field tensor and a transformed electric field tensor to an update generation module;
generating an updated transformed electric field tensor at the update generation module according to the standard FDTD process, using the transformed permittivity tensor, the transformed electric displacement field tensor, and the transformed electric field tensor;
communicating the updated transformed electric field tensor to the transformation module; and
applying an inverse unitary transformation at the processor of the transformation module to the updated transformed electric field tensor to generate an updated initial electric field tensor in the initial reference frame; and
using the updated electric field tensor in the initial reference frame in an update of the standard FDTD process while running the simulation.
2. The computer-implemented method according to claim 1 , wherein applying the unitary transformation further comprises applying the unitary transformation at the transformation module to an initial electric field tensor in the initial reference frame to generate the transformed electric field tensor in the transformed reference frame.
3. The computer-implemented method according to claim 1 , wherein applying the inverse unitary transformation further comprises applying an inverse unitary transformation at the transformation module to the transformed electric displacement field tensor.
4. The computer-implemented method according to claim 1 , wherein the medium is anisotropic and the transformed reference frame is aligned with an anisotropic axis of the medium.
5. The computer-implemented method according to claim 1 , wherein the initial reference frame is a global coordinate system.
6. The computer-implemented method according to claim 1 , wherein the transformed reference frame is a local coordinate system.
7. The computer-implemented method according to claim 1 , wherein each mesh element is a Yee cell.
8. A computer-implemented method for simulating an electromagnetic property of a medium, the medium being meshed to define a plurality of mesh elements in an initial reference frame, the method comprising, for each mesh element:
determining a unitary transformation at a transformation module having a processor configured for transforming an initial permeability tensor in the initial reference frame into a transformed permeability tensor in a transformed reference frame, the transformed permeability tensor being diagonal;
using a standard finite-difference time-domain (FDTD) process for simulating the electromagnetic property of the medium;
while modifying the standard FDTD process by:
applying the unitary transformation at the transformation module to an initial magnetic flux density tensor in the initial reference frame to generate a transformed magnetic flux density tensor in the transformed reference frame;
communicating the transformed permeability tensor, the transformed magnetic flux density tensor and a transformed magnetic field tensor to an update generation module;
generating an updated transformed magnetic field tensor at the update generation module according to the standard FDTD process, using the transformed permeability tensor, the transformed magnetic flux density tensor, and the transformed magnetic field tensor;
communicating the updated transformed magnetic field tensor to the transformation module; and
applying an inverse unitary transformation at the processor of the transformation module to the updated transformed magnetic field tensor to generate an updated initial magnetic field tensor in the initial reference frame; and
using the updated magnetic field tensor in the initial reference frame in an update of the standard FDTD process while running the simulation.
9. The computer-implemented method according to claim 8 , wherein applying the unitary transformation further comprises applying the unitary transformation at the transformation module to an initial magnetic field tensor in the initial reference frame to generate the transformed magnetic field tensor in the transformed reference frame.
10. The computer-implemented method according to claim 8 , wherein applying the inverse unitary transformation further comprises applying an inverse unitary transformation at the transformation module to the transformed magnetic flux density tensor.
11. The computer-implemented method according to claim 8 , wherein the medium is anisotropic and the transformed reference frame is aligned with an anisotropic axis of the medium.
12. The computer-implemented method according to claim 8 , wherein the initial reference frame is a global coordinate system.
13. The computer-implemented method according to claim 8 , wherein the transformed reference frame is a local coordinate system.
14. The computer-implemented method according to claim 8 , wherein each mesh element is a Yee cell.
15. A non-transitory computer readable medium having stored thereon computer readable instructions for causing a processor to simulate an electromagnetic property of a medium, the medium being meshed to define a plurality of mesh elements in an initial reference frame, the computer readable medium comprising instructions for:
determining a unitary transformation at a transformation module having a processor configured for transforming an initial permittivity tensor in the initial reference frame into a transformed permittivity tensor in a transformed reference frame, the transformed permittivity tensor being diagonal;
using a standard finite-difference time-domain (FDTD) process for simulating the electromagnetic property of the medium;
while modifying the standard FDTD process by:
applying the unitary transformation at the transformation module to an initial electric displacement field tensor in the initial reference frame to generate a transformed electric displacement field tensor in the transformed reference frame;
communicating the transformed permittivity tensor, the transformed electric displacement field tensor and a transformed electric field tensor to an update generation module;
generating an updated transformed electric field tensor at the update generation module according to the standard FDTD process, using the transformed permittivity tensor, the transformed electric displacement field tensor, and the transformed electric field tensor;
communicating the updated transformed electric field tensor to the transformation module; and
applying an inverse unitary transformation at the processor of the transformation module to the updated transformed electric field tensor to generate an updated initial electric field tensor in the initial reference frame; and
using the updated electric field tensor in the initial reference frame in an update of the standard FDTD process while running the simulation.
16. A non-transitory computer readable medium having stored thereon computer readable instructions for causing a processor to simulate an electromagnetic property of a medium, the medium being meshed to define a plurality of mesh elements in an initial reference frame, the computer readable medium comprising instructions for:
determining a unitary transformation at a transformation module having a processor configured for transforming an initial permeability tensor in the initial reference frame into a transformed permeability tensor in a transformed reference frame, the transformed permeability tensor being diagonal;
using a standard finite-difference time-domain (FDTD) process for simulating the electromagnetic property of the medium;
while modifying the standard FDTD process by:
applying the unitary transformation at the transformation module to an initial magnetic flux density tensor in the initial reference frame to generate a transformed magnetic flux density tensor in the transformed reference frame;
communicating the transformed permeability tensor, the transformed magnetic flux density tensor and a transformed magnetic field tensor to an update generation module;
generating an updated transformed magnetic field tensor at the update generation module according to the standard FDTD process, using the transformed permeability tensor, the transformed magnetic flux density tensor, and the transformed magnetic field tensor;
communicating the updated transformed magnetic field tensor to the transformation module; and
applying an inverse unitary transformation at the processor of the transformation module to the updated transformed magnetic field tensor to generate an updated initial magnetic field tensor in the initial reference frame; and
using the updated magnetic field tensor in the initial reference frame in an update of the standard FDTD process while running the simulation.