EP3008619A2 - Computer simulation of electromagnetic fields - Google Patents
Computer simulation of electromagnetic fieldsInfo
- Publication number
- EP3008619A2 EP3008619A2 EP14810283.3A EP14810283A EP3008619A2 EP 3008619 A2 EP3008619 A2 EP 3008619A2 EP 14810283 A EP14810283 A EP 14810283A EP 3008619 A2 EP3008619 A2 EP 3008619A2
- Authority
- EP
- European Patent Office
- Prior art keywords
- boundary
- potential
- function
- basis function
- basis
- Prior art date
- Legal status (The legal status is an assumption and is not a legal conclusion. Google has not performed a legal analysis and makes no representation as to the accuracy of the status listed.)
- Withdrawn
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Classifications
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- G—PHYSICS
- G06—COMPUTING; CALCULATING OR COUNTING
- G06F—ELECTRIC DIGITAL DATA PROCESSING
- G06F17/00—Digital computing or data processing equipment or methods, specially adapted for specific functions
- G06F17/10—Complex mathematical operations
- G06F17/11—Complex mathematical operations for solving equations, e.g. nonlinear equations, general mathematical optimization problems
- G06F17/13—Differential equations
-
- G—PHYSICS
- G01—MEASURING; TESTING
- G01R—MEASURING ELECTRIC VARIABLES; MEASURING MAGNETIC VARIABLES
- G01R33/00—Arrangements or instruments for measuring magnetic variables
- G01R33/0064—Arrangements or instruments for measuring magnetic variables comprising means for performing simulations, e.g. of the magnetic variable to be measured
-
- G—PHYSICS
- G06—COMPUTING; CALCULATING OR COUNTING
- G06F—ELECTRIC DIGITAL DATA PROCESSING
- G06F30/00—Computer-aided design [CAD]
- G06F30/30—Circuit design
- G06F30/36—Circuit design at the analogue level
-
- G—PHYSICS
- G01—MEASURING; TESTING
- G01R—MEASURING ELECTRIC VARIABLES; MEASURING MAGNETIC VARIABLES
- G01R29/00—Arrangements for measuring or indicating electric quantities not covered by groups G01R19/00 - G01R27/00
- G01R29/08—Measuring electromagnetic field characteristics
-
- G—PHYSICS
- G06—COMPUTING; CALCULATING OR COUNTING
- G06F—ELECTRIC DIGITAL DATA PROCESSING
- G06F2111/00—Details relating to CAD techniques
- G06F2111/10—Numerical modelling
Definitions
- a parameterized solution to Maxwell's equations is written (for example, a sum of parameters times plane waves, or a sum of parameters times fields due to fictitious "equivalent sources").
- the parameterization and the parameters are chosen so that the error is small in some sense.
- FIG. 1 shows a block diagram of a machine that may be used for carrying out the computations for determining electromagnetic fields.
- FIG. 2 is a block diagram of an example of system, which may store the machine code for solving for electromagnetic fields.
- FIG. 3 is a flowchart of an embodiment of a method that is implemented by processor system.
- FIG. 4 is a flowchart of an embodiment of a method for solving the electromagnetic equations.
- FIG. 5 shows an example of homogeneous regions for computing the electromagnetic fields.
- FIG. 6 shows an example of computational regions corresponding to the homogeneous regions of example with a thin boundary region.
- [1012JFIG. 7 shows is a diagram illustrating elements of an example to which the least action method is applied.
- FIG. 8 shows another sample problem having a dielectric sphere and a point dipole.
- FIG. 9 shows an example of a placement of virtual sources inside the sphere of FIG. 8 for computing the basis potential functions outside of the sphere.
- FIG. 10 shows an example of a placement of virtual sources outside the sphere of FIG. 8 for computing basis function within the sphere.
- FIG. 11 shows an example of an arrangement of virtual point dipoles on the polyhedron of FIG. 9.
- FIG. 12 shows an example of using a mesh of simplices for computing a field.
- each of FIGs. 1, 2, 5-11 is a brief description of each element, which may have no more than the name of each of the elements in the one of FIGs. 1, 2, 5-11 that is being discussed. After the brief description of each element, each element is further discussed in numerical order. In general, each of FIGs. 1-11 is discussed in numerical order and the elements within FIGs. 1-11 are also usually discussed in numerical order to facilitate easily locating the discussion of a particular element. Nonetheless, there is no one location where all of the information of any element of FIGs. 1-11 is necessarily located. Unique information about any particular element or any other aspect of any of FIGs. 1-11 may be found in, or implied by, any part of the specification.
- FIG. 1 shows a block diagram of a machine 100 that may be used for carrying out the computations for determining electromagnetic fields.
- Machine 100 may include output system 102, input system 104, memory system 106, processor system 108, communications system 112, and input/output device 114. In other embodiments, machine 100 may include additional components and/or may not include all of the components listed above.
- Machine 100 is an example of what may be used for carrying out the computations for determining electromagnetic fields.
- Machine 100 may be a computer and/or a special purpose machine computing electromagnetic computations.
- Output system 102 may include any one of, some of, any combination of, or all of a monitor system, a handheld display system, a printer system, a speaker system, a connection or interface system to a sound system, an interface system to peripheral devices and/or a connection and/or interface system to a computer system, intranet, and/or internet, for example. Output system 102 may be used to display indication the types of input needed, whether there are any issues with the inputs received, and output results of a computation of an
- electromagnetic field for example.
- Input system 104 may include any one of, some of, any combination of, or all of a keyboard system, a mouse system, a track ball system, a track pad system, buttons on a handheld system, a scanner system, a microphone system, a connection to a sound system, and/or a connection and/or interface system to a computer system, intranet, and/or internet (e.g., IrDA, USB), for example.
- Input system 104 may be used to enter the problem parameters, such as the grid on which to perform the computations, the material parameters of each region within the grid, choose basis functions, and initial state, if applicable.
- Memory system 106 may include, for example, any one of, some of, any combination of, or all of a long term storage system, such as a hard drive; a short term storage system, such as random access memory; a removable storage system, such as a floppy drive or a removable drive; and/or flash memory.
- Memory system 106 may include one or more machine-readable mediums that may store a variety of different types of information.
- the term machine-readable medium is used to refer to any non-transient medium capable carrying information that is readable by a machine.
- One example of a machine -readable medium is a computer-readable medium.
- Memory system 106 may store the machine instructions (e.g., a computer program, which may be referred to as a code) that cause machine 100 to compute electromagnetic field computations.
- Memory 106 may also store material parameters corresponding to specific materials and basis-functions.
- the basis function may be parameterized solutions to the electromagnetic equations within a homogenous region.
- Processor system 108 may include any one of, some of, any combination of, or all of multiple parallel processors, a single processor, a system of processors having one or more central processors and/or one or more specialized processors dedicated to specific tasks.
- Processor system 110 may carry out the machine instructions stored in memory system 108, and compute the electromagnetic field.
- Communications system 112 communicatively links output system 102, input system 104, memory system 106, processor system 108, and/or input/output system 114 to each other.
- Communications system 112 may include any one of, some of, any combination of, or all of electrical cables, fiber optic cables, and/or means of sending signals through air or water (e.g. wireless communications), or the like.
- Some examples of means of sending signals through air and/or water include systems for transmitting electromagnetic waves such as infrared and/or radio waves and/or systems for sending sound waves.
- Input/output system 114 may include devices that have the dual function as input and output devices.
- input/output system 114 may include one or more touch sensitive screens, which display an image and therefore are an output device and accept input when the screens are pressed by a finger or stylus, for example.
- the touch sensitive screens may be sensitive to heat and/or pressure.
- One or more of the input/output devices may be sensitive to a voltage or current produced by a stylus, for example.
- Input/output system 114 is optional, and may be used in addition to or in place of output system 102 and/or input device 104.
- FIG. 2 is a block diagram of an example of system 200, which may store the machine code for solving for electromagnetic fields.
- System 200 may include action extremizer 201 and user interface 202 having tolerance choices 204, grid choices 206, output choices 208, basis function choices 210, and material parameter choices 212.
- Memory 106 may also include basis functions 214, initial state generator 218, and output generator 220.
- system 200 may include additional components and/or may not include all of the components listed above.
- system 200 is an embodiment of memory system 106, and the blocks of system 200 represent different functions of the code that solves for electromagnetic fields performs, which may be different modules, units and/or portions of the code.
- system 200 may be a portion of the processor system, such as an Application Specific Integrated Circuit (ASIC), and/or another piece of hardware in which the code of system 200 is hardwired into system 100.
- ASIC Application Specific Integrated Circuit
- a frame vector of the 3D space of the inertial frame associated with ⁇ 0 is represented by 3 ⁇ 4 ⁇ ⁇ for i E ⁇ 1,2,3 ⁇ .
- electromagnetic field F is defined to equal the antisymmetric part of the space-time vector derivative of the space-time vector potential A
- ⁇ is the outer product of geometric algebra
- ⁇ 0 is the time-like space-time velocity vector defining the inertial frame associated with the 3D space vector potential A and scalar potential ⁇
- quantities with an over-script arrow such as A are always vectors in the 3D space of the inertial frame defined by ⁇ 0 .
- the word “potential” is generic to the space- time vector potential, to the scalar and vector potential in 3D space together or separately, to the "media potential” and/or to other simply related quantities including a discontinuous potential that is related to a continuous potential by a gauge transformation.
- the electromagnetic field F is related to the electric field E and "magnetic B -field" or “magnetic intensity" B by
- III ⁇ 3 ⁇ 43 ⁇ 43 ⁇ 4 YoYiY 2 Y3 is the pseudoscalar of the geometric algebra of both 3D space and 4D space-time.
- the methods described herein are not limited to linearly polarizable material, for linearly polarizable media the field G is the sum of F and a polarization field P,
- the word "field” is generic to F or G, or their parts E, B, D, or H, or the simply related media field F m .
- the polarizations P and M are generally modeled as functions of E and B, often linear functions, but not necessarily.
- the space-time vector field J representing charge-current density in 4D space-time volumes is related to the charge density p and 3D vector current density J by the equation and the space-time vector field K representing charge-current density on 3D boundaries in 4D space-time is related by the equation
- Extremizer 201 extremizes the action integral by adjusting coefficients of a sum of basis. After the coefficients have been adjusted by the extremizer, the sum of basis functions is a solution or an approximation of a solution to the electromagnetic fields.
- User interface 202 may be used for receiving input for setting up the problem to solve and outputting results of computing electromagnetic fields.
- Tolerance choice 204 is an input for entering the tolerance for performing the computations or another form of input for determining the accuracy with which to perform the computations.
- Grid choices 206 includes one or more input fields for defining the space in which the problem is solved, including a description of boundaries between regions with different polarizabilities. Grid choices 206 may also include input fields for defining the initial state of the system. The grid choices 206 divide the computation region into two or more smaller computation regions or cells.
- a grid may be chosen to divide space or space-time into computation regions that are simplices, or to divide the boundary between regions into simplices.
- a simplex (which in plural form is simplexes or simplices) refers to a generalization of the notion of a triangle or tetrahedron to arbitrary dimension.
- an n-simplex is an n-dimensional polytope, which is the convex hull of n + 1 vertices.
- a 2-simplex is a triangle
- a 3-simplex is a tetrahedron
- a 4-simplex is a pentachoron.
- Simplices may be desirable when the parameterized potential is defined to be a linear function of position within each simplex.
- a continuous potential may be parameterized by the values of the continuous potential at all simplex vertices, and the field F is a constant within each simplex because the field F is defined to be a linear combination of derivatives of A.
- the field G is a constant within each simplex if the field G is a function of only the field F, whether G is a linear or nonlinear function of F.
- Simplices may be defined that cover all of the simulation space, or may be combined with regions of other or arbitrary shape. Including simplices with or between regions of arbitrary shape may make it easier to define good potential basis functions.
- Basis functions 210 may include a set of default basis functions that the user can choose, such as those specified by boundary values and calculated with the help of Green's theorem as an integral of the boundaries of each simplex.
- Basis functions 210 may also include an input charge-current source or associated basis field.
- Basis function choices 210 may include a list of default basis functions, which the user may choose.
- the default basis functions when multiplied by a series of quantities (which may be referred to as parameters) and summed together, may form parameterized functions, and solving for the parameters solves the electromagnetic equations.
- the basis functions 210 may include potential basis functions and field basis functions. In this specification, the term basis function is generic to potential basis functions, field basis functions, and other basis functions.
- each potential basis function may be specified by potential values on boundaries along with the requirement that at non-boundary points the Euler-Lagrange equation is satisfied by the potential for any set of parameter values.
- each potential basis function may be specified by values of the scalar potential and tangential values of the vector potential on boundaries and the requirement that the potential satisfies the Lorenz condition at region points next to a boundary, along with the requirement that at non-boundary points the potential satisfies the wave equation in the media for any set of parameter values.
- Basis functions may be defined in terms of fields that would result from fictitious "virtual" charge and current densities, such as point dipole moments, in a corresponding fictitious space filled with an unbounded uniform polarizable medium.
- one or more virtual point dipole moments at each of a series of points are used to define each basis function within each region with uniform polarizability.
- a good set of basis functions is a set basis functions capable of representing any function that may represent the electromagnetic fields in the non-boundary regions.
- a good set of basis functions is a complete set of basis functions.
- the virtual dipole moments and/or other virtual charge and current densities are located outside of a computation cell and/or region in which the basis function is being defined.
- Material parameters 212 may include the material parameters that the user can choose for each region of the space in which the problem is being solved, for example.
- material parameters may include combinations of material parameters associated with common materials, so that the user can just specify the name of the material and the combination of parameters associated with that material may be assigned to the region chosen.
- Basis function 214 may include an algorithm for choosing basis functions when the user has not chosen and/or may include an algorithm for converting the user's input into a function that is called by the code as the basis functions that are used by least action extremizer 201 to extremize the action integral and solve for the fields.
- Action extremizer 201 is a module that solves for the electromagnetic fields by extremizing the action.
- action extremizer 201 computes numerical values for a set of parameters of basis functions (e.g., the vector and scalar potentials from which the electric and magnetic fields may be derived).
- the potential functions are a parameterized continuous electromagnetic potential.
- the values of the parameters are computed within a predetermined tolerance to extremize an action integral.
- the action integral is the integral of the Lagrangian density.
- the Lagrangian density contains terms for the electromagnetic field in vacuum or in polarizable media and for the interaction of the electromagnetic field with electric charge and current.
- the terms of the electromagnetic fields may describe linear or nonlinear polarizable media.
- a method for accurately and efficiently computing values of electromagnetic fields, suitable for use in various fields of applied physics and engineering ranging from magnetostatics to RF engineering to integrated optics.
- An electromagnetic potential is defined that is parameterized by the electromagnetic potential's tangential components at boundaries between regions with uniform polarizability, and parameter values are chosen that extremize the classical electromagnetic action.
- Useful general results are derived using the geometric algebra of space-time. The method is compared with the widely used method of moments.
- the current section contains a summary of the principle of least action as applied to electromagnetism in space-time, using the notation of and results from [1 ,2], and SI units. For simplicity the problem is restricted to linearly polarizable media.
- the electromagnetic action S with a space-time vector charge-current source density J is a scalar- valued functional S [A, VA] of a continuous space-time vector field A and derivatives VA of A, over all of space-time volume u, given by the integral where the electromagnetic field F is a space-time bivector field defined as a derivative of the space-time potential A, a space-time vector field, by
- P P— IM is the electric vector plus magnetic bivector
- equation (5) After writing F in terms of A and using linear dependence of G on F, the first term of equation (5) can be integrated by parts so that equation (5) can be written as
- Equation (6) is true if and only if the Euler-Lagrange equation
- V G - J f 0 (7) holds throughout space-time u.
- the space-time bivector field F may be projected into two parts, one part containing and one part not containing a unit time-like vector ⁇ 0 associated with any particular inertial reference frame, as
- the first term of equation (9) is a 3D space vector proportional to the electric field E and the second is a 3D space bivector proportional to the magnetic intensity B.
- the space-time bivector field G may be expressed as the sum of parts projected onto and rejected by y 0 ,
- V D p f (13)
- V X H - ⁇ C (14)
- Discontinuities in polarizability of the medium with change in position in space correspond to discontinuities in permittivity ⁇ or permeability ⁇ .
- the space-time potential A is continuous everywhere by assumption, discontinuity in polarizability generally corresponds to discontinuities in derivatives of A and therefore in F and in G.
- space-time is filled with bounded regions of 4D volume having continuous polarizability with discontinuities only at the boundaries between the bounded regions, then such discontinuities in F and G occur only at the boundary points.
- the collection of boundary points be called ⁇ and the collection of region points be called u and let the corresponding surface charge-current density and spatial charge-current density be called K f and J f , respectively. Then the equation from which the Euler-Lagrange equation was deduced can, in the instant case, be written as
- any variation ⁇ in the field A is considered. But in numerical computations, typically the field A is parameterized with a finite number of parameters c a that may range over all possible values or may be constrained to range over some specified set of values
- Each basis field a a is defined and continuous at every point of space-time, including boundary points.
- a Greek index such as a ranges over all values while a Roman index such as i ranges over i ⁇ 0 is used herein.
- Equation (19) is a general result that may be applied by the method of least action to compute electromagnetic fields. Specifically, equation (19) applies to both linear and nonlinear media, although for simplicity the discussion is continued below for linear media.
- F is a linear function of A and G is a linear function of F
- F and G can each be written as corresponding sums of basis functions
- the general expressions for the matrix and array elements for frequency ⁇ are identical to those of (26) and (27), but with d t replaced by - io) and integration done over only the volume V and boundary s, and not time t.
- Other diverging factors such as ( ⁇ )— ( ⁇ ) () , representing a physical resonance with numerical problems at frequency ⁇ 0 , can be similarly eliminated.
- V ⁇ G J f
- V ⁇ G J f
- V - gi 0 (29) at all non-boundary points.
- the volume integrals in the matrix elements given by (24) and (25), or equivalently by (27) and (28), are all identically zero.
- each potential basis function a a may be defined by specifying the value of potential basis function a a at all points on boundaries between homogeneous regions, and by defining the function within each homogenous region to satisfy the
- V m ( £ ⁇ + (3 ⁇ 4 (1) ⁇ ) ⁇ 0 (34)
- d t is replaced by - io
- the Euler-Lagrange equation may be rewritten in terms of these quantities as
- ) + V - A 0 - either are equally valid gauges to which the electromagnetic equations may be constrained to remove the gauge invariance, thereby simplifying solving the electromagnetic equations numerically.
- the Euler-Lagrange equation simplifies to the wave equation
- a useful method of defining a space-time potential in a bounded region such as a potential basis function
- [1087]4) use the gauge condition and Green's Theorem on the side of each of the boundary points that is within the region (in which the current computation is being performed), as constraints to reduce the number of degrees of freedom at each boundary point from 8 to 3. It may be convenient to choose the 3 remaining degrees of freedom to be the components of the potential that are tangential to the boundary in space-time. Specifically, the 4 space-time components of the potential at every boundary point plus the 4 space-time components of the normal derivative of the potential at every boundary point gives 8 degrees of freedom. Next, Greens Theorem is applied, which gives 4 equations (or 4 degrees of constraint) for every boundary point that relates all 8 of the degrees of freedom reducing the number of degrees of freedom to 4.
- the Lorenz condition provides one more equation (or 1 more degree of constraint), which relates only derivatives and further reduces the number of degrees of freedom to 3.
- the 3 chosen degrees of freedom (e.g., the tangential components of the potential function) at each boundary point can be given by a finite number of parameters in a model of the potential, from which all 8 degrees of freedom of the potential and its normal derivative, and therefore the fields f a and g a , can be quickly calculated at all points on the boundary.
- the potential and field value at any point in the region may then be calculated from the values on the boundary using Greens Theorem [3].
- the boundary between regions may be modeled or approximated as a set of connected triangles, with the potential specified at each vertex by 3 parameters and defined to vary linearly with position at all other points on each triangle.
- Each such basis function may be chosen to have nonzero tangential boundary components on only one localized part of the boundary, such as by choosing the parameters of the basis function to be nonzero at only one vertex. Choosing nonzero tangential boundary components on only one localized part of the boundary results in the integrands of the boundary integrals of (19) being zero everywhere except in the localized part of the boundary having the nonzero tangential boundary components, since the integrand is independent of the
- V -T 0 describes the conservation of energy and momentum at all non- boundary points
- J ⁇ ⁇ (n ⁇ ⁇ ) ⁇ describes a potential- weighted boundary integral conservation law for energy and momentum at boundary points.
- J ⁇ ⁇ (n ⁇ ⁇ ) ⁇ ensures that the integral or average of energy momentum over a small patch identified by the localized basis function a j - but not at every infinitesimal point - is exactly conserved across the boundaries between regions of computation.
- a and A' have the same tangential boundary values, result in the same field F, and differ by only a gauge transformation; we may say such potentials are "gauge-equivalent".
- the current section discusses the calculation of potentials and fields at boundary points. Specifically, the current section describes a method for calculating all components of the potential, and calculating the normal directional derivative of the potential at all boundary points enclosing a region with uniform polarizability, given the boundary conditions of only the tangential components of the potential. The electromagnetic field at all boundary points can then be calculated from the components of the potential and the normal directional derivative of the potential, which allows evaluation of the boundary integrals needed to extremize the action.
- the first step is to specify the value of each basis potential a a and the normal derivative of the basis potential a a at boundary points by a) requiring that each basis potential a a satisfies the corresponding homogenous or inhomogeneous Euler-Lagrange equation and the media Lorenz condition, and therefore the media wave equation, at all points enclosed by the given boundary, b) giving the values of each basis potential a a tangential components at all boundary points, and c) using the media Lorenz condition and Greens Theorem for the media wave equation as constraints to calculate the basis potential normal component and all components of the normal derivative of the basis potential at boundary points.
- each electromagnetic field basis function g a must be found on each side of every boundary at every boundary point.
- the method described in the current section enables the efficient calculation of the normal component of a space-time vector potential a and the normal directional derivatives of all components of a at all boundary points, given the tangential components of a, for a potential that satisfies the Euler- Lagrange equation and the Lorenz condition such as any one of the basis potentials a a .
- the value of any field basis function g a can be calculated from the boundary values and normal derivatives of the potential basis function a a .
- the potential function a and the potential function's normal derivatives at all boundary points are defined in terms of a finite number of variables.
- the potential a may represent, for example, any one of the basis functions a a .
- the following simple model illustrates the method, although the same method can be used to define smoother albeit more complex models, or less smooth models.
- a general curved boundary is approximated by connected boundary simplices.
- the boundary simplexes are triangles and it is useful to visualize the instant case as representative.
- the unit vector normal to the surface of the simplex s may be written as n s .
- n s is specified, which may be normal to the physical boundary that is being approximated by the connected simplices, which may be defined as the normalized sum of the area vectors of the simplices that share that vertex.
- the resulting potential solution depends only weakly on the choice of normal vectors n v , so the exact definition is not critical.
- the derivative in the direction n v of the component of a that is parallel to n v is determined from the values of a on the boundary by the medium Lorenz condition. At any one vertex, this derivative in the direction n v of the component of a that is parallel to n v is generally different for each triangle that shares that vertex. All 4 components of the potential and all 4 components of the normal derivative of the potential at all points on the boundary are in expressed in terms of only 7 parameters at each vertex.
- the tangential components of the potential are continuous across any boundary, and so have the same values as the boundary is approached from either side, but the normal component of the potential and all components of the directional derivative of the potential are generally different on the two sides of a boundary between media with different polarizabilities; each side must be treated independently.
- the potential a at any point on a boundary simplex is defined to be the linear function of position on that simplex that matches the vertex values a v .
- the media potential a m is just equal to a linear transformation of the components of a at each point, so it is possible to use either the function a or a m as the potential (the form of the resulting equations are unchanged) and transform from one to the other as needed.
- Green's theorem is applied to eliminate 4 of the 7 degrees of freedom in the parameters a v (with 4 degrees) and n v ⁇ Vn v ii v Aa v (with 3 degrees) at each vertex v, as follows.
- Equations (42) and (43) apply if the medium's polarizability is zero. If the medium's polarizability is not zero, then a similar expression but with a, j f , V, and k replaced by the media quantities a m , j m , V m , k m , as described earlier, is used.
- These p space-time vector equations make 4p scalar equations of constraint (where p is the number of vertex points).
- constraints can be used to write all 7p parameters as linear functions of the 3p parameters n v ii v Aa v (i.e., the components of the potential a that are perpendicular to the vertex normal vector n v ) at any boundary vertex v.
- the Lorenz or media Lorenz condition can then be applied at any boundary point to also calculate the 8th degree of freedom (the normal derivative of the normal component of the potential) at any boundary point.
- Each basis function aj is chosen to be described by setting one component of the vertex parameter n v ii v Aa v at one vertex equal to unity, and all other vertex parameters and the source j f to zero.
- the basis function a 0 may be chosen to be described by setting the 3p vertex parameters at each vertex to equal zero but setting the source j f to equal the specified source J f , or J m if using media quantities, of the simulation.
- the normal derivative of the potential component perpendicular to the simplex at any point on any boundary simplex can be numerically evaluated from these values.
- each integral needed to extremize the action has a non-zero integrand over only a localized region of the boundary for which the tangential components of the corresponding basis function are non-zero.
- These integrals using localized basis potentials can be much faster to compute than integrals using non- localized basis potentials.
- Green's theorem expression from the previous section can be used with the final potential solution A in place of a.
- ⁇ [ ⁇ ] (-ik + ) ⁇ [?] where k ⁇ — .
- F can be numerically calculated from values of A at nearby points.
- Each potential and field may be parameterized as a sum of parameters C j times corresponding basis functions.
- the method of least action identifies parameter values that extremize the action.
- the action may be extremized subject to a constraint equation such as the preceding example by Lagrange's method. Such constraints may be enforced by defining an auxiliary action function S' given by
- Virtual sources may be used to define a basis potential a j within any one particular region R.
- the virtual charge-current source ) is chosen to be nonzero only at points not in region R. That is, the virtual charge- current source jf* is located at points on the boundary or outside of the region.
- the potential may be chosen to satisfy the wave equation and the Lorenz condition.
- the virtual charge- current source is parameterized, and the parameters are chosen so that the resulting potential meets the boundary conditions (e.g., the specified scalar potential and tangential vector potential values on the boundary) within an acceptable error.
- Parameterizing the virtual charge-current source results in an expression for the basis potential a j at any point f in region R that is an acceptably good approximation, within region R, for the original problem involving a space with boundaries between regions of uniform polarizability. Every region is parameterized, and then the parameterization process is repeated for every basis potential.
- the resulting potential can satisfy the boundary conditions to a good approximation and satisfy the wave equation and the Lorenz condition exactly.
- the virtual source for each basis potential and each region may be a collection of point electric and magnetic dipoles with well-chosen locations, and with parameterized amplitudes chosen to satisfy the boundary conditions within a good
- both the potential and the electromagnetic field associated with a given virtual source can be quickly calculated using simple well-known expressions for the potential and field due to dipoles.
- the basis potential a j can be chosen to yield the correct field to an approximation that is as good as desired.
- a useful measure of error in meeting the boundary conditions is the integral over the boundary of
- the potential may be chosen as the product of the potential value on the boundary times a function of the perpendicular distance from the boundary chosen so that the resulting potential approximately or exactly satisfies the Euler-Lagrange equation.
- the function of the perpendicular distance might, for example, be found as or approximated by a power series, an exponential function, or the product of a power series and exponential function.
- the medium potential A m is generally used and transforms to the potential A after finding a desired solution.
- Computing A m can be a useful method, because the differential equation for A m is simple. Assuming for simplicity that there is no specified charge or current J m inside the region of interest, the differential equation for A m is
- V m 2 A m 0 .
- k m
- /v m is the medium wavenumber.
- the solution to the preceding equation may be written as an exponential function with a generally complex argument. Other approximations and solutions may also be used.
- the region may contain a boundary of discontinuity in polarizability, in which case the potential approximation may be continuous but piecewise linear across the discontinuity.
- the potential A may be computed, and then the fields F and then G may be computed on the boundary of the region. Assuming the potential A and field G have already been determined on the other side of the boundary, the change AG in the field G across the boundary can be calculated. Using the calculation of AG, the boundary integrals required for extremizing the action may also be calculated.
- FIG. 3 is a flowchart of an embodiment of a method 300 that is implemented by processor system 300.
- a determination is made of how to set up the grid, such as the size of the different regions of the problem area.
- material parameters are chosen for each region.
- a determination is made as to what basis functions to use for solving the electromagnetic fields.
- the basis functions are parameterized solution to the electromagnetic equations in a homogeneous region.
- the action is extremized (e.g., minimized), which may involve adjusting the parameters of the parameterized basis functions to extremize the action integral at the boundary between homogenous regions.
- the output of action extremizer 201 is formatted and presented to the user.
- each of the steps of method 300 is a distinct step.
- step 302-310 may not be distinct steps.
- method 300 may not have all of the above steps and/or may have other steps in addition to or instead of those listed above.
- the steps of method 300 may be performed in another order. Subsets of the steps listed above as part of method 300 may be used to form their own method.
- FIG. 4 is a flowchart of an embodiment of a machine implemented method 400 for solving the electromagnetic equations.
- Method 400 may be an embodiment, of step 308 of method 300.
- step 402 the region in which the problem is being solved is divided into a series of homogenous regions, each homogeneous region being a region in which electromagnetic properties of the material such as permittivity, permeability, and electrical conductivity are homogeneous throughout the region.
- the machine may choose the dimensions of the matrices and the number of iterations in the loops in the computer code that correspond to the representing the fields and parameters of the equations that are appropriate for the chosen regions.
- step 404 choose a parameterized potential function that satisfies the Euler-Lagrange electromagnetic equations in the homogeneous regions but not necessarily on the boundaries between regions, which results in the volume integrals that appear in the equation for extremizing the action to be identically zero, leaving only boundary integrals to be calculated, and choose the functions to satisfy a gauge condition to simplify subsequent calculations
- the Euler-Lagrange electromagnetic equations are typically written using tensor algebra or geometric algebra of space-time and are equivalent to Maxwell's inhomogeneous equations - the two equations without charge density or current - which are typically written using vector algebra of 3D space.
- the other two of Maxwell's equations are equivalent to a mathematical identity when written using tensor or geometric algebra of space-time.
- a practical choice of such a parameterized potential function is a sum of terms, each term equal to a parameter times a basis function.
- the potential is further characterized by choosing how the potential is parameterized (tangential components of the potential values on the boundary in space-time) and calculating an expression that gives other necessary values (the non-tangential component, and all components of the normal derivative) in terms of the parameters (the tangential components of the potential in space-time are equal to the tangential components of the 3D space vector potential and the scalar potential).
- the user may enter a choice of basis functions and/or the machine may automatically choose the basis functions depending on the homogeneous regions chosen in step 402.
- step 406 the equations to be satisfied by the parameters are identified by writing them as symbolic equations appropriate for choices made so far.
- the machine may automatically write/determine the equations based on the choices made so far.
- the user may be offered a choice of equations to solve and/or may be provided with a field for entering the equation that the user desires to solve.
- each of these equations contains a term for each parameter, multiplied typically by one boundary integral.
- step 407 for any one of these equations that has a potentially diverging factor of 1/f where f may be very small or zero, as will typically occur if the problem contains an electrical conductor that is represented by a complex -value permittivity, the equation is first multiplied by f and simplified symbolically to cancel any factors of 1/f before numerical evaluation of coefficients of parameters in the equation.
- Step 407 may be performed as a result of user input making entering the choice or the code may automatically identify the 1/f dependence and choose the equations resulting from the multiplication by f as the equation to solve.
- the equations are solved by the computer for numerical values of the parameters.
- each of the steps of method 400 is a distinct step.
- step 402-406 may not be distinct steps.
- method 400 may not have all of the above steps and/or may have other steps in addition to or instead of those listed above.
- the steps of method 400 may be performed in another order. Subsets of the steps listed above as part of method 400 may be used to form their own method.
- FIG. 5 shows an example 500 of homogeneous regions.
- Example 500 includes region 502, region 504, and boundary 506.
- One region may have a finite extent and be completely surrounded by the other, and one region may extend to infinity in all directions and completely surround the other.
- example 500 may include additional components and/or may not include all of the components listed above.
- Region 502 is homogeneous region 1, which has a homogeneous set of material parameters. In other words, throughout region 502 each of the material parameters is assigned a uniform value.
- the material parameters may include the permittivity (or susceptibility), permeability, polarization, magnetization, resistivity (or conductivity), for example. Parameter may also be included that specify the charge density and current density. A specified electric polarization or magnetization may be represented by the corresponding specified bound charge or bound current densities.
- Region 504 is homogeneous region 2, which also has a
- Boundary 506 separates region 502 and region 504, and in an embodiment, is where the computations are primarily formed.
- Region 502 and region 502 may represent two physical regions that have distinctly different sets of material parameters with the same values as assigned during the computations.
- a region having a continuously varying set of material parameters may be modeled in various ways, such as by dividing the region into smaller regions of homogeneous material parameters.
- FIG. 6 shows an example 600 of computational regions corresponding to the homogeneous regions of example 500.
- Example 600 includes region 602, region 604, and boundary 606.
- system 600 may include additional components and/or may not include all of the components listed above.
- Region 602 is an interior region of homogeneous region 1
- region 604 is an interior region of homogeneous region 2.
- region points may generally be considered to exclude boundary points, but the phrase "interior region point” or “non-boundary point” may be used to make it explicitly clear that the point is in the region and not a boundary point if it is important in a discussion that the point is not a boundary point.
- region 602 and region 604 no actual computations may need to be carried out.
- Each potential basis function extends over all of space and the potential basis function, or a potential related by a gauge transformation to the potential basis function, is continuous at all points of space, which in the example of FIG. 6 includes regions 602 and 604.
- Each such continuous potential function typically has a "kink" across a boundary, corresponding to discontinuity in the derivative of the function.
- the values of any given basis function aj in either or both region 602 and region 604 may be defined with the help of virtual sources, although different virtual sources will be used for each region and the material parameters, such as polarizability, of each region may differ.
- Boundary 606 includes points representing a boundary between two regions.
- Boundaries are modeled as mathematical surfaces, which are usually smooth.
- a real physical boundary may be very close to a mathematical surface, such as the boundary between a glass and air, or may have some small, but negligible, thickness, such as the boundary between a piece of wood and adjacent concrete.
- the computation of the electromagnetic fields by extremizing the action integrals are primarily performed using integrals over the points of boundary 606, allowing the computational parameters Cj to be chosen to extremize the action.
- region is generally used to refer to a volume containing material with uniform polarizability since solutions to the wave equation are well known in the instant case, but regions with non-uniform polarizability may also be used, especially if solutions are known. Also, although the polarizability may be discontinuous across the boundary between regions, the polarizability is not necessarily discontinuous. For example, a volume with uniform polarizability since solutions to the wave equation are well known in the instant case, but regions with non-uniform polarizability may also be used, especially if solutions are known. Also, although the polarizability may be discontinuous across the boundary between regions, the polarizability is not necessarily discontinuous. For example, a volume with uniform
- polarizability can be divided into two regions, joined by a boundary across which the polarizability does not change. Dividing the volumes into two regions can be convenient for various reasons. For example, dividing the volume into two regions allows the use of different methods of defining basis functions in these two adjacent regions for faster or more accurate computations.
- Volumes in 4D space-time may be denoted by u, and 3D boundaries in 4D space-time may be denoted by ⁇ .
- Volumes in 3D space may be denoted by V, and 2D boundaries in 3D space may be denoted by s.
- FIG.7 shows a diagram illustrating elements of an example 700 to which the method of least action is applied.
- Example 700 includes medium 702, medium 704, interface 706, z-axis 708, and origin 710.
- a person familiar with electromagnetism may also use the ideas of incident wave 712, travel direction 714, transmitted wave 716, travel direction 718, reflected wave 720, and travel direction 722.
- the ideas of Fig.7 are not needed with the method using Green's theorem, but the ideas of Fig.7 are identified here to help explain the method.
- medium 702 is the medium on the right side of the diagram and the functions and quantities associated within medium 702 have the subscript R.
- Medium 702 has a permittivity 8R and a permeability of ⁇ .
- medium 704 is the medium on the left side of the diagram and the functions and quantities associated within medium 704 have the subscript L.
- Medium 704 has a permittivity £ L and a permeability of ⁇ ⁇ .
- Interface 706 is the interface between the two media. In the coordinate system of FIG.
- z-axis 708 indicates the position of the waves traveling within mediums 702 and 704.
- Origin 710 is the point on z-axis 708 at which the value of the position coordinate is 0.
- Incident wave 712 is a plane wave of light entering from the right hand side of FIG. 7.
- the source of the incident wave is modeled as a sheet of oscillating current perpendicular to the z axis.
- the source of incident wave 712 and the material parameters of media 702 and 704 are the inputs to the problem.
- the incident wave (that would be due to the source if the incident wave were not interacting with the system being simulated) may alternatively be an input instead of the source of the wave, but in the current example the simpler method of using the source as input is applied.
- Travel direction 714 is the direction in which incident wave 712 travels.
- reflected wave 720 is a wave of light that was reflected from interface 706 as a result of incident wave 712 hitting interface 706.
- Travel direction 722 is the direction in which reflected wave 720 travels, which is the opposite direction as the travel direction 714 of incident wave 712.
- transmitted wave 716 is a wave of light that was transmitted through interface 706 as a result of incident wave 712 hitting interface 706.
- Travel direction 718 is the direction in which transmitted wave 716 travels, which is the same direction as incident wave
- the current example is done using Green's theorem methods.
- Green's theorem the basis functions satisfy the wave equation and the Lorenz condition in media at all non- boundary points, so that the volume integrals in the action are zero.
- V a. -— iooa; H V ⁇ a.
- Greens' theorem can be inverted to find the normal derivative of the potential on the boundary as a function of the values of the potential on the boundary.
- the general 3D result as applied to a complex 3D vector potential a * j may be written
- FIG. 8 shows a sample problem 800.
- Sample problem 800 includes dipole 802, polarizable sphere 804.
- Dipole 802 has a moment pointing in the direction of the arrow representing the dipole.
- Polarizable sphere 804 is made from a homogeneous polarizable material and the space outside of the sphere is nonpolarizable.
- the surface of the sphere 804 is approximated by connected triangles.
- the objective of problem 800 is to solve for the electric and magnetic fields induced by dipole 802 inside and outside of the sphere. The problem will not be completely solved here, but the calculation of basis functions using two methods that may be used to solve the problem, Green's theorem and virtual sources, will be discussed.
- Space may be divided into two regions for the purposes of computing the
- One region may be the interior of the sphere 804, the other region may be everything outside of sphere 804, and the surface of the sphere is the boundary between these two regions.
- Other methods of defining basis functions may additionally divide either or both of these two regions (the region including the inside of sphere 804 and the region including everywhere else) with homogeneous polarizability into two or more regions.
- a general method for defining a set of basis functions using either Green's Theorem or virtual sources is to first i) for each basis function, specify the values of the tangential components of the space-time vector potential field at all boundary points between
- each boundary vertex For each boundary vertex, define a vector ii that is normal to the physical boundary (such as, for example, an average of the normal vectors of the triangles that share that vertex), and choose three basis potential functions: one with nonzero scalar potential, and two with specified vector potential values perpendicular to the normal 3D space vector at that vertex (the normal vector component is one of the unknowns that is found).
- step ii) can be done using Green's theorem following the recipe previously outlined.
- virtual sources for each basis function, choose a set of virtual sources inside the sphere to define the basis function outside the sphere such that the discrepancy between the potential calculated from the virtual sources and the specified boundary values for the basis potential is minimized, or more accurately reduced to within an acceptable tolerance.
- a set of virtual sources may be chosen outside the sphere to define the basis function inside the sphere.
- Each virtual source may be a
- the virtual sources may be chosen to define a basis potential, the same virtual sources may be used to calculate the corresponding electric and magnetic field F or G.
- the potential inside the sphere may be defined as a linear combination of potentials due to virtual sources located outside the sphere. For convenience a linear combination of the same set of virtual sources may be chosen for each basis function, but for each basis function the
- coefficients of the combination is chosen that gives the smallest integrated error relative to the specified values of the basis function on the boundary. Note that these coefficients are in addition to the parameters Cj used to extremize the action.
- FIG. 9 shows an example 900 of a placement of virtual sources inside the sphere.
- Example 900 shows sphere 804 with a polyhedron 902 inside having virtual dipoles 904 at the vertices of the polyhedron that is inside the spherical boundary.
- Polyhedron 902 is used to aid in locating the dipoles 904, which are placed close to the inner surface of the sphere 802.
- Dipoles 904 are used to generate the basis potentials aj outside the sphere. The potential in the outer region from each dipole 904 in the inner region is used as a different basis function a j having a unit dipole moment.
- the basis potential aj is defined inside the sphere as a linear combination of potentials due to virtual dipoles located outside the sphere (which are discussed further in conjunction with FIG. 10, below), with the coefficients chosen to minimize (reduce to within a predetermined acceptable tolerance) the effective discontinuity of the basis potential aj across the surface of the sphere.
- a possible measure of effective discontinuity is described in another section.
- the coefficients of the linear combination used to define the basis function for each basis index i are determined early in the solution by minimizing the discontinuity at boundary points, and are in addition to the parameters C j that are later chosen to extremize the action.
- FIG. 10 shows an example 1000 of a placement of virtual sources outside the sphere.
- Example 1000 shows sphere 802 with a polyhedron 1002 outside having virtual dipoles 1004 at the vertices of the polyhedron.
- Polyhedron 1002 is used to aid in locating the dipoles 1004, which are placed close to the outer surface of the sphere 802.
- Dipoles 1004 are used to generate the basis potentials a j inside the sphere. For each basis function a i ?
- the potential a j is defined inside the sphere by a linear combination of the potential resulting from dipoles 1004 outside the sphere, with the amplitudes of the dipoles chosen to minimize the discontinuity in the basis potential at the boundary.
- FIG. 11 shows an example 1100 of an arrangement of dipoles on the polyhedron.
- Example 1100 shows polyhedron 902 with virtual dipoles 904 at the vertices of the polyhedron and electric dipole 1102, magnetic dipole 1104, and magnetic dipole 1106.
- Electric dipole 1102 points towards the surface of the sphere and is perpendicular to the surface of the sphere.
- Magnetic dipoles 1104 and 1106 are perpendicular to the electric dipole 1102 and are therefore parallel to the surface of the sphere at the point on the sphere to which the electric dipole points.
- one pair of virtual dipoles - one inside and one outside the sphere - at the correct locations can be used to define the exact solution for the fields of the simple ideal case of a real dipole located outside of a polarizable sphere.
- the exact solution method of solving for the field resulting from a charged object outside of the shape does not apply and may be difficult to implement even approximately, while the method of least action with basis functions defined with the help of virtual sources may be applied with good results.
- point dipole virtual sources are very useful for representing and visualizing geometry, and have very simple and easily calculated potentials and fields, for smoother fields, smoother virtual charge and current densities may be used instead of point dipoles.
- FIG. 12 shows an example 1200 of a mesh of simplices that may be used computing electromagnetic fields.
- Example 1200 includes wires 1204 and 1206, and simplices 1208 having nodes 1210 and boundaries 1212.
- FIG. 12 shows a two-dimension problem, which was chosen because the situation of FIG. 12 can be graphically represented in 2D.
- FIG. 12 there is a metal object that varies in the x and y directions (in the plane of the paper), and extends very far in the z direction (perpendicular to the paper) so that the situation may approximate a 2D problem, with all quantities
- Wires 1204 have a current that travels out of the paper in the left (which are represented by large grey dots), wires 1206 have a current that travels into of the paper on the left (which are represented by large black dots). Wires 1204 and 1206 may represent a coil that is stretched in the z direction.
- Simplices 1208 are an example of a mesh of simplices that may be used for performing the field computations.
- Nodes 1210 are the vertexes of the simplices, and boundaries 1212 are the walls between the simplices (connecting the vertexes).
- the size of the simplices may be varied according to the amount of variation expected in a particular region. Areas with higher variation in the field may include a larger number of simplices, but the simplices will tend to be smaller than other areas, and areas with less variation in the fields may be have a smaller number of simplices, but the simplices will tend to be larger.
- parameterization to be linear in the parameters. If a parameterization is chosen that is nonlinear in the parameters, a solution may be found that extremizes the action by any of various other methods, such as Newtonian iteration with linearized approximations to the potentials and fields.
- the fields G, D, and H are linear functions of the fields F, E, and B, respectively, but these may be nonlinear functions in which case the Lagrangian density may need additional corresponding terms and solutions may be found that extremize the action by any of various methods, such as using perturbation theory and choosing a set of basis functions that includes a fundamental frequency and harmonics.
- An example of a nonlinear field G is an electrically and magnetically polarizable medium with an electric polarization P that saturates according to
- least action method requires that the potential be continuous at all points, including boundary points, and so computation by the method of least action is most easily analyzed for errors and understood if the potential is continuous at all points. But the potential may be discontinuous in at least two acceptable ways and the method still be useful. This is detailed below.
- the value of ⁇ is uniquely determined at every point of a 4D space- time volume, if either ⁇ or the directional derivative of ⁇ perpendicular to the boundary is specified on the 3D boundary of the 4D volume.
- a potential that is continuous except for a discontinuous perpendicular vector part at a boundary is therefore effectively continuous and is not a problem.
- a basis potential may be discontinuous
- the imperfect matching of basis potential values at points on the boundary generally occurs with the method of virtual sources but not with the Green's Theorem method.
- the system and method may be used to design the shape, placement, and materials for magnetic recording write heads, antenna, the core of a transformer, the core of electromagnets, permanent magnets, and/or electromagnets for generators, and/or transmission lines, electric conductors and electronic components in larger assemblies such as computers and cell phones, microwave devices, and optical devices, including optical and electro-optical integrated circuits.
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US14/261,415 US9286419B2 (en) | 2010-07-19 | 2014-04-24 | Computer simulation of electromagnetic fields |
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