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US 9,286,419 B2Granted
Computer simulation of electromagnetic fields
AssigneeTerje Graham Vold
InventorTerje Graham Vold
Life of this patent
Estimating the term…
Abstract
A method and system is provided for solving for electromagnetic fields by approximating an electromagnetic function as a sum of basis functions multiplied by coefficients to be determined. The set of equations used to determine the coefficients results from taking derivatives of the action integral with respect to the coefficients (and/or other parameters) and setting the derivative equal to zero, thereby extremizing the action integral.
Claims45
A machine implemented method comprising:
- computing, by a machine having a processor system including one or more processors and a memory system, numerical values for a set of parameters that extremize an action integral, the set of parameters are parameters of a parameterized electromagnetic potential and an electromagnetic field associated with the parameterized electromagnetic potential, the computing of the numerical values for the set of parameters that extremize the action integral is performed, therein numerically solving for the parameterized electromagnetic potential by computing the numerical values of the parameters, instead of solving Euler-Lagrange equations for the electromagnetic potential in at least part of a region associated with the action integral and instead of solving Euler-Lagrange equations for the electromagnetic fields in at least part of a region associated with the action integral,
- the set of parameters, computed by the machine, extremizing an action integral, the action integral being an integral of an electromagnetic Lagrangian;
- and
- computing, by the machine, the electromagnetic field based on the set of parameters that extremize the action integral.
1 more element
A machine implemented method comprising:
- computing, by a machine having a processor system including one or more processors and a memory system, numerical values for a set of parameters that extremize an action integral for a parameterized electromagnetic potential and an electromagnetic field associated with the parameterized electromagnetic potential, the set of parameters, computed by the machine, extremizing an action integral, the action integral being an integral of an electromagnetic Lagrangian;
- and
- computing, by the machine, the electromagnetic field based on the set of parameters that extremize the action integral,
- the parameterized electromagnetic potential being a linear combination of basis functions that satisfy an Euler-Lagrange equation for non-boundary points, and
- as a result of the parameterized electromagnetic potential being the linear combination of basis functions that satisfy the Euler-Lagrange equation for non-boundary points, the action integral including contributions only from integrals including boundary points.
2 more elements
A machine implemented method comprising:
- computing, by a machine having a processor system including one or more processors and a memory system, numerical values for a set of parameters that extremize an action integral for a parameterized electromagnetic potential and an electromagnetic field associated with the parameterized electromagnetic potential, the set of parameters, computed by the machine, extremizing an action integral, the action integral being an integral of an electromagnetic Lagrangian;
- and
- computing, by the machine, the electromagnetic field based on the set of parameters that extremize the action integral,
- wherein time-dependent quantities vary harmonically with time at one frequency, and
- wherein if an equation that results from extremizing the action integral has a factor of 1/f, where f is a function of frequency that may be zero for at least one value of frequency, then this equation is multiplied by a factor off and simplified symbolically to cancel the factor of 1/f before the full set of equations is solved numerically for the set of parameters.
2 more elements
A machine implemented method comprising:
- computing, by a machine having a processor system including one or more processors and a memory system, numerical values for a set of parameters that extremize an action integral for a parameterized electromagnetic potential and an electromagnetic field associated with the parameterized electromagnetic potential, the set of parameters, computed by the machine, extremizing an action integral, the action integral being an integral of an electromagnetic Lagrangian;
- computing, by the machine, the electromagnetic field based on the set of parameters that extremize the action integral;
- and
- constraining the potential to satisfy an equation that defines a gauge of the potential, as part of the computing, by the machine, of the numerical values for the set of parameters.
1 more element
A machine implemented method comprising:
- computing, by a machine having a processor system including one or more processors and a memory system, numerical values for a set of parameters that extremize an action integral for a parameterized electromagnetic potential and an electromagnetic field associated with the parameterized electromagnetic potential, the set of parameters, computed by the machine, extremizing an action integral, the action integral being an integral of an electromagnetic Lagrangian;
- computing, by the machine, the electromagnetic field based on the set of parameters that extremize the action integral;
- and
- parameterizing the electromagnetic potential to form the parameterized electromagnetic potential by requiring, during the computing by the machine, that the electromagnetic potential agree with a parameterized potential function on boundaries between regions of computation, the potential function satisfying a Euler-Lagrange equation at non-boundary points.
1 more element
A machine implemented method comprising:
- computing, by a machine having a processor system including one or more processors and a memory system, numerical values for a set of parameters that extremize an action integral for a parameterized electromagnetic potential and an electromagnetic field associated with the parameterized electromagnetic potential, the set of parameters, computed by the machine, extremizing an action integral, the action integral being an integral of an electromagnetic Lagrangian;
- computing, by the machine, the electromagnetic field based on the set of parameters that extremize the action integral, the computing of the set of parameters including at least solving a set of equations, by the machine, in a plurality of regions within which the polarizability is differentiable with respect to position and separated by boundaries, the potential being a linear combination of one or more basis potentials, with at least one basis function being defined in at least one region as a potential function of at least one or more virtual charge-current sources, the virtual charge-current source being located outside the region.
A machine implemented method comprising:
- computing, by a machine having a processor system including one or more processors and a memory system, numerical values for a set of parameters that extremize an action integral for a parameterized electromagnetic potential and an electromagnetic field associated with the parameterized electromagnetic potential, the set of parameters, computed by the machine, extremizing an action integral, the action integral being an integral of an electromagnetic Lagrangian;
- computing, by the machine, the electromagnetic field based on the set of parameters that extremize the action integral;
- the computing of the set of parameters including at least solving, by the machine, a set of equations in a plurality of regions within which the polarizability is differentiable with respect to position and separated by boundaries on which the polarizability is not differentiable, the potential being a linear combination of one or more basis potentials, with at least one basis function, being defined in at least one region as a potential function of at least one or more virtual charge-current sources, the virtual charge-current source being located outside the region.
A machine implemented method comprising:
- computing, by a machine having a processor system including one or more processors and a memory system, numerical values for a set of parameters that extremize an action integral for a parameterized electromagnetic potential and an electromagnetic field associated with the parameterized electromagnetic potential, the set of parameters, computed by the machine, extremizing an action integral, the action integral being an integral of an electromagnetic Lagrangian;
- computing, by the machine, the electromagnetic field based on the set of parameters that extremize the action integral;
- the computing of the numerical values including at least solving, by the machine, a set of equations in a plurality of regions, the potential being a linear combination of one or more basis potentials, with at least one basis function being defined in at least one region as a potential function of:
- at least one virtual charge-current sources, the at least one virtual charge-current source being located outside the region, and
- at least on real charged current sources located inside the region.
2 more elements
A machine implemented method comprising:
- computing, by a machine having a processor system including one or more processors and a memory system, numerical values for a set of parameters that extremize an action integral for a parameterized electromagnetic potential and an electromagnetic field associated with the parameterized electromagnetic potential, the set of parameters, computed by the machine, extremizing an action integral, the action integral being an integral of an electromagnetic Lagrangian;
- computing, by the machine, the electromagnetic field based on the set of parameters that extremize the action integral;
- solving, by the machine, a set of equations in a plurality of regions including one or more simplices, and
- the parameterized potential in one or more simplices satisfying the Euler-Lagrange equations within a tolerance.
1 more element
A machine implemented method comprising:
- computing, by a machine having a processor system including one or more processors and a memory system, numerical values for a set of parameters that extremize an action integral for a parameterized electromagnetic potential and an electromagnetic field associated with the parameterized electromagnetic potential, the set of parameters, computed by the machine, extremizing an action integral, the action integral being an integral of an electromagnetic Lagrangian;
- computing, by the machine, the electromagnetic field based on the set of parameters that extremize the action integral;
- wherein the extremizing includes solving, by the machine, a set of equations
- ∂
- c
- i
-
- ∫
- (
- 1
- 2
-
- F
- ·
- G
- -
- A
- ·
- J
- f
- )
-
- ⅆ
- u
- =
- 0
- where
- ∫ indicates to perform an integration,
- u is a four dimensional volume having three spatial dimensions and one time dimension over which the integration is performed,
- F is an electromagnetic field strength,
- A is the electromagnetic potential,
- J is free or total charge-current density,
- U is the interaction between the field and polarizable media,
- ci are the parameters of the potential A such that A≈Σiaici;
- and are parameters of the electromagnetic field F such that F≈Σifici
- where ai and fi are basis functions of A and F, respectively,
- such that ai is a function of space-time position, if ai is not a continuous function of space-time position, any discontinuities present are removable by a gauge transformation,
- and fi is a function of space-time position given by fi=∇{circumflex over ( )}ai.
35 more elements
A system comprising:
- a processor system having one or more processors;
- and
- a memory system including a one or more non-transitory machine readable media;
- the non-transitory machine readable media storing one or more machine instructions, which when implemented by the processor system cause the processor system to perform a method including at least
- computing, by the machine, numerical values for a set of parameters of a parameterized electromagnetic potential, by the processor system choosing parameter values that extremize an action integral whose integrand includes at least a Lagrangian density with terms for at least an electromagnetic field, instead of solving Euler-Lagrange equations for the electromagnetic potential in at least part of a region associated with the action integral and instead of solving Euler-Lagrange equations for the electromagnetic fields in at least part of a region associated with the action integral.
2 more elements
A non-transitory computer readable medium storing one or more machine instruction, which when implemented by a processor causes the processor to perform a method comprising:
- computing, by a machine having a processor system including one or more processors and a memory system, numerical values for a set of parameters of a parameterized electromagnetic potential by the processor system extremizing an action integral within a predetermined tolerance for where the action integral is an integral of an electromagnetic Lagrangian density, the extremizing being performed by the machine at least numerically solving a set of equations for the set of parameters, the set of equations being equations that result from taking partial derivatives of the action integral with respect to the set of parameters.
A system comprising a processor system having one or more processors and a memory system storing one or more machine instructions, which when invoked cause the system to implement a method comprising:
- computing, by the processor system including one or more processors and a memory system, numerical values for a set of parameters of an action integral;
- the computing of the numerical values being performed instead of solving Euler-Lagrange equations associated with the action integral in at least part of a region associated with the action integral, the set of parameters being the parameters of an integrand of the action integral,
- the set of parameters, computed by the processor system, extremizing an action integral, the action integral being an integral of an electromagnetic Lagrangian;
- and
- computing, by the machine, electromagnetic quantities indicative of electromagnetic behavior of a system characterized by the electromagnetic Lagrangian;
- the computing of the electromagnetic quantities, being based on the set of parameters that extremize the action integral.
3 more elements
A machine implemented method comprising:
- computing, by a machine having a processor system including one or more processors and a memory system, numerical values for a set of parameters of an integrand of an action integral, instead of solving Euler-Lagrange equations associated with the action integral in at least part of a region associated with the action integral,
- the numerical values computed by the machine for the set of parameters extremizing the action integral, and the action integral being an integral of an electromagnetic Lagrangian;
- and
- computing, by the machine, electromagnetic quantities indicative of electromagnetic behavior of a system characterized by the electromagnetic Lagrangian;
- the computing of the electromagnetic quantities, being based on the set of parameters that extremize the action integral.
2 more elements
A system comprising a processor system having one or more processors and a memory system storing one or more machine instructions, which when invoked cause the system to implement a method comprising:
- computing, by a machine having a processor system including one or more processors and a memory system, numerical values for a set of parameters that are coefficients of basis functions of an integrand of an action integral, the numerical values being computed from values of one or more integrals that are part of the action integral;
- the numerical values computed by the machine for the set of parameters extremize the action integral, and the action integral being an integral of an electromagnetic Lagrangian;
- and
- computing, by the machine, electromagnetic quantities indicative of electromagnetic behavior of a system characterized by the electromagnetic Lagrangian;
- the computing of the electromagnetic quantities, being based on the set of parameters that extremize the action integral.
2 more elements
A machine implemented method comprising:
- computing, by a machine having a processor system including one or more processors and a memory system, numerical values for a set of parameters of an integrand of an action integral, the numerical values being computed from values of one or more integrals that are part of the action integral;
- the numerical values computed by the machine for the set of parameters extremizing the action integral, and the action integral being an integral of an electromagnetic Lagrangian;
- and
- computing, by the machine, electromagnetic quantities indicative of electromagnetic behavior of a system characterized by the electromagnetic Lagrangian;
- the computing of the electromagnetic quantities, being based on the set of parameters that extremize the action integral.
2 more elements
Dependencies are inferred from the claim text.