Simulations of physical systems for multiple excitations
Physical systems modeled by matrix equations may simultaneously be simulated for multiple excitations using, for example, an adaptive Krylov iterative recycling method.
1. A computer-implemented method of electronically simulating a physical system simultaneously for a plurality of excitations, the method comprising:
(a) storing, in computer memory, a model of the physical system based on a system matrix; and
(b) computationally simulating, using the computer, the physical system by:
(1) initiating approximations of physical quantities associated with the physical system for the plurality of excitations; and
(2) iteratively improving the approximations based, at least in part, on the model by:
(i) computing a plurality of residual errors, each residual error being based on a residual vector corresponding to a different one of the plurality of excitations;
(ii) selecting an excitation from the plurality of excitations corresponding to a maximum one of the plurality of residual errors;
(iii) performing a single conjugate gradient residual (CGR) iteration based on the excitation corresponding to the maximum residual error;
(iv) updating the approximations for all of the excitations based, at least in part, on the selected excitation; and
(v) repeating substeps (i) through (iv) until a termination condition is satisfied.
2. The method of claim 1 , wherein the physical system comprises a physical field in a user-defined domain.
3. The method of claim 2 , wherein the physical quantities comprise components of the physical field at a plurality of locations within the user-defined domain.
4. The method of claim 2 , wherein the physical field comprises an electromagnetic field.
5. The method of claim 1 , wherein the physical system comprises a physical network.
6. The method of claim 5 , wherein the physical quantities are associated with components of the physical network.
7. The method of claim 5 , wherein the physical network comprises at least one of an electronic circuit, an electrical circuit, a pneumatic circuit, and a hydraulic circuit.
8. The method of claim 1 , wherein updating the approximations for all of the excitations comprises computing a Krylov vector for each iteration based, at least in part, on the selected excitation, and updating the approximations and the residual errors associated therewith for all of the plurality of excitations based, at least in part, on the Krylov vector for the respective iteration.
9. The method of claim 8 , wherein updating the approximations for all of the excitations further comprises enforcing that the Krylov vectors are linearly independent.
10. The method of claim 9 , wherein enforcing the linear independence of the Krylov vectors comprises conjugating the Krylov vectors by modified Gram-Schmidt conjugation.
11. The method of claim 1 , wherein initiating the approximations of the physical quantities comprises setting the physical quantities to zero.
12. The method of claim 1 , wherein each residual vector comprises a difference between the corresponding excitation and the corresponding approximation multiplied by the system matrix.
13. The method of claim 1 , wherein simulating the physical system further comprises aborting the iterative improvements when the residual errors fall below a user-defined threshold.
14. The method of claim 12 , wherein each residual error comprises a ratio of a norm of the corresponding residual vector and a norm of a corresponding excitation vector.
15. A computer system for simulating a physical system simultaneously for a plurality of excitations, the computer system comprising:
(a) a memory for storing parameters representative of the physical system; and
(b) an analysis module for modeling the physical system with a system matrix;
(c) a simulation module for simulating the physical system by
(1) initiating approximations of physical quantities associated with the physical system for the plurality of excitations; and
(2) iteratively improving the approximations based, at least in part, on the model by
(i) computing a plurality of residual errors, each residual error being based on a residual vector corresponding to a different one of the plurality of excitations;
(ii) selecting an excitation from the plurality of excitations corresponding to a maximum one of the plurality of residual errors;
(iii) performing a single conjugate gradient residual (CGR) iteration based on the excitation corresponding to the maximum residual error;
(iv) updating the approximations for all of the excitations based, at least in part, on the selected excitation; and
(v) repeating substeps (i) through (iv) until a termination condition is satisfied.
16. The computer system of claim 15 , wherein the physical system comprises a physical field in a user-defined domain.
17. The computer system of claim 15 , wherein the analysis module updates the approximations for all of the excitations by computing a Krylov vector for each iteration based, at least in part, on the selected excitation, and updating the approximations and the residual errors associated therewith for all of the plurality of excitations based, at least in part, on the Krylov vector for the respective iteration.