CN112967401A - Three-dimensional space electric field calculation method based on finite element method - Google Patents
Three-dimensional space electric field calculation method based on finite element method Download PDFInfo
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Abstract
The invention relates to the technical field of electric field research, in particular to a method for calculating a three-dimensional space electric field based on a finite element method. The method comprises the steps of carrying out three-dimensional modeling and space simulation on a three-dimensional space electric field, carrying out calculation analysis on an electric field boundary value, carrying out calculation analysis on a minimum electrostatic energy boundary value problem, subdividing a simulation model, carrying out calculation analysis on a unit energy value, carrying out multivariate quadratic function extreme value calculation, calculating an electric field intensity vector and the like. The invention designs a computer engineering software which adopts a finite element method as a principle to carry out simulation modeling of a three-dimensional space electric field, disperses the simulation electric field into a plurality of units to carry out energy calculation, obtains a more accurate approximate result of the electric field intensity by calculating the approximate solution of each unit, realizes the intensity analysis of a virtual electric field, and further not only can provide theoretical basis and support for the line structure design of a power transmission line, but also can provide a technical method for inhibiting the electrostatic induction and transient electric shock influence of the built line.
Description
Technical Field
The invention relates to the technical field of electric field research, in particular to a method for calculating a three-dimensional space electric field based on a finite element method.
Background
The electric shock phenomenon suffered by a human body in the area near the circuit can be divided into a steady-state electric shock and a transient-state electric shock according to different acting time, wherein the transient-state electric shock phenomenon is more common. Common transient electric shocks are easy to occur in metal structures and metal pipelines in the area near the line, and scenes such as clothes hangers, umbrella opening by residents, automobile parking and the like. Because no corresponding environmental protection standard evaluation exists, the treatment of the inductive electricity dispute event of the power transmission line close to the civil house is difficult and troublesome, and is difficult to properly solve. Therefore, the system is required to develop researches on evaluation and suppression measures of transient electric shocks below the overhead transmission line under different voltage classes and different scenes, reveal the electrostatic induction phenomenon and the action process of the electrostatic induction phenomenon with the human body, analyze the feeling characteristics of the human body suffering from the transient electric shocks under different electrostatic induction voltages, and provide measures for suppressing and alleviating the influence of the induction voltages and the transient electric shocks under different scenes on the production and life of residents. However, since the concept of the electric field is virtual, and there is a great risk in performing experimental studies in an actual electric field, the electric field strength can be studied by establishing a three-dimensional electric field. However, the three-dimensional space electric field simulated by adopting computer engineering software has the problem that the electric field intensity is difficult to accurately calculate and control. Therefore, if a more accurate approximate value of the electric field intensity can be calculated, a better research effect can be achieved. The results obtained from the finite element analysis calculation are only approximate, but if the number of the divided elements is very large and reasonable, the obtained results are consistent with the actual situation. At present, there is no perfect method for calculating and analyzing the three-dimensional space electric field by a finite ternary method.
Disclosure of Invention
The invention aims to provide a method for calculating a three-dimensional space electric field based on a finite element method, so as to solve the problems in the background technology.
In order to solve the above technical problem, an object of the present invention is to provide a method for calculating a three-dimensional space electric field based on a finite element method, including the steps of:
s1, selecting COMSOL software based on a finite element method as a principle, and performing three-dimensional modeling and space simulation on the three-dimensional space electric field;
s2, calculating and analyzing the electric field boundary value of the simulated three-dimensional electric field space;
s3, calculating and analyzing the minimum electrostatic energy boundary value problem;
s4, subdividing the simulation model, and calculating and analyzing the unit energy value;
s5, discretizing the functional problem, then performing multivariate quadratic function extremum calculation, and calculating the approximate value required by each node;
and S6, calculating an electric field intensity vector by combining the potential distributions.
As a further improvement of the present technical solution, in S2, the calculation expression of the electric field boundary value is:
in the electrostatic field, the electric field intensity vector and the electric displacement vector meet the loop theorem and the Gaussian theorem, rho is the charge density of the free body, and equation (1) and equation (2) are differential equations used for electrostatic field analysis in the Maxwell equation set;
will be provided with
Substituting formula (2) and considering D ═ E, one can obtain:
in the formula (3), ε is a dielectric constant, and an identity is used
Equation (3) can be expressed as:
the formula (4) is a function of potential
And bulk charge density p, dielectric constant epsilon is constant in a region of uniform medium,
equation (4) can be simplified to:
the expression of equation (5) in the rectangular coordinate system is:
in the region where ρ is 0, there are:
equation (6) is called the poisson equation of the electrostatic field, and equation (7) is called the laplace equation of the electrostatic field, where the laplace equation is a special case of the poisson equation.
As a further improvement of the present technical solution, in S3, the minimum electrostatic energy margin has three types of problems, and the expression thereof is:
the electrostatic field minimum action principle is adopted, a fixed belt system is positioned in a medium, the surface charge distribution of the fixed belt system enables the synthesized electric field to have minimum electrostatic energy, and the expression of the first-class boundary value problem energy integral is as follows:
the electrostatic field problem is therefore equivalent to the functional extremum problem:
similarly, the equivalence functional extreme value problems of the second and third types of edge value problems are as follows:
therein, functional
Quadratic dependence function
And partial derivatives thereof, hence the term
Is a function of
The corresponding variation problem is called a quadratic functional extreme value problem.
As a further improvement of the present technical solution, in S4, the method for performing calculation analysis on the unit energy value includes the following steps:
s4.1, performing secondary functional calculation of all unit energies;
s4.2, sequentially carrying out energy integral calculation of the single unit;
s4.3, calculating unit energy integrals which are independent of x, y and z respectively;
and S4.4, simultaneously combining the equations of all the sub-regions.
As a further improvement of the present technical solution, in S4.1, performing a quadratic functional calculation on all unit energies may be expressed as a sum of all unit energy functions:
as a further improvement of the present technical solution, in S4.2, an expression for performing single-unit energy integral calculation is as follows:
and the following steps:
thus:
namely:
as a further improvement of the present technical solution, in S4.3, expressions for calculating the unit energy integrals independent of x, independent of y, and independent of z are respectively:
if it is
Independent of x, then:
wherein:
wherein S isεThe area enclosed by the subdivision area;
in the same way, if
Independently of y, then:
wherein S isεThe area enclosed by the subdivision area;
in the same way, if
Independent of z, then:
wherein S isεThe area enclosed by the subdivision area.
As a further improvement of the present technical solution, in S4.4, an expression obtained by combining equations in a simultaneous manner is as follows:
from equation (12) -equation (22), it can be derived:
by combining equations (23) for all sub-regions concurrently, equation (11) can be written as:
wherein
And K is the matrix of variables and coefficients after all the equation sets are connected.
As a further improvement of the present technical solution, in S5, the expression of discretizing the functional problem and then performing multivariate quadratic function extremum calculation is as follows:
from the theory of extreme values of functions, there are:
the formula is as follows:
wherein, the formula (27) is a finite element equation, and the equation (27) is solved in sequence to obtain the node of each node
The values may be used to obtain the desired approximate solution.
As a further improvement of the present technical solution, in S6, the expression for calculating the electric field strength vector is as follows:
the electric field intensity vector E of each potential can be obtained by combining the potential distribution.
Another object of the present invention is to provide an apparatus for calculating a three-dimensional space electric field based on a finite element method, including a processor, a memory, and a computer program stored in the memory and executed on the processor, wherein the processor is configured to implement any of the steps of the method for calculating a three-dimensional space electric field based on a finite element method when the computer program is executed.
It is a further object of the present invention to provide a computer-readable storage medium storing a computer program, which when executed by a processor, implements the steps of any of the above-described methods for calculating a three-dimensional space electric field based on a finite element method.
Compared with the prior art, the invention has the beneficial effects that: in the method for calculating the three-dimensional space electric field based on the finite element method, computer engineering software which takes the finite element method as a principle is adopted to carry out simulation modeling on the three-dimensional space electric field, the finite element method is adopted to calculate the boundary value of the simulation electric field and the minimum electrostatic energy of the electric field, the simulation electric field is dispersed into a plurality of units to carry out energy calculation, and an accurate electric field strength approximate result is obtained by calculating the approximate solution of each unit to realize the intensity analysis of the virtual electric field, so that not only can theoretical basis and support be provided for the circuit structure design of the power transmission line, but also a technical method can be provided for the electrostatic induction and transient electric shock influence suppression of the built circuit, and the method has great significance for building an environment-friendly power grid.
Drawings
FIG. 1 is an overall process flow diagram of the present invention;
FIG. 2 is a flow chart of a partial method of the present invention;
FIG. 3 is a block diagram of an exemplary computer program product of the present invention.
Detailed Description
The technical solutions in the embodiments of the present invention will be clearly and completely described below with reference to the drawings in the embodiments of the present invention, and it is obvious that the described embodiments are only a part of the embodiments of the present invention, and not all of the embodiments. All other embodiments, which can be derived by a person skilled in the art from the embodiments given herein without making any creative effort, shall fall within the protection scope of the present invention.
Method embodiment
As shown in fig. 1 to fig. 3, the present embodiment aims to provide a method for calculating a three-dimensional space electric field based on a finite element method, which includes the following steps:
s1, selecting COMSOL software based on a finite element method as a principle, and performing three-dimensional modeling and space simulation on the three-dimensional space electric field;
s2, calculating and analyzing the electric field boundary value of the simulated three-dimensional electric field space;
s3, calculating and analyzing the minimum electrostatic energy boundary value problem;
s4, subdividing the simulation model, and calculating and analyzing the unit energy value;
s5, discretizing the functional problem, then performing multivariate quadratic function extremum calculation, and calculating the approximate value required by each node;
and S6, calculating an electric field intensity vector by combining the potential distributions.
Specifically, the finite element method is a numerical calculation method for solving mathematical equations, is a numerical analysis technology organically combining elastic theory, computational mathematics and computer software, and is a powerful numerical calculation tool for solving the practical engineering problem.
The finite element method is expressed in a matrix form, and is convenient for compiling a computer program. At present, computer engineering software which can be conveniently and directly applied at home and abroad comprises MSC NASTRAN, ADINA, LS-DYNA, ANSYS, ANSOFT, ABAQUS, 2D-sigma, COMSOL and the like, wherein the COMSOL and ANSYS are both suitable for electric field analysis.
In this embodiment, in S2, the calculation expression of the electric field boundary value is:
in the electrostatic field, the electric field intensity vector and the electric displacement vector meet the loop theorem and the Gaussian theorem, rho is the charge density of the free body, and equation (1) and equation (2) are differential equations used for electrostatic field analysis in the Maxwell equation set;
will be provided with
Substituting formula (2) and considering D ═ E, one can obtain:
in the formula (3), ε is a dielectric constant, and an identity is used
Equation (3) can be expressed as:
the formula (4) is a function of potential
And bulk charge density p, dielectric constant epsilon is constant in a region of uniform medium,
equation (4) can be simplified to:
the expression of equation (5) in the rectangular coordinate system is:
in the region where ρ is 0, there are:
equation (6) is called the poisson equation of the electrostatic field, and equation (7) is called the laplace equation of the electrostatic field, where the laplace equation is a special case of the poisson equation.
In this embodiment, in S3, the minimum electrostatic energy margin has three types of problems, and the expression thereof is:
the electrostatic field minimum action principle is adopted, a fixed belt system is positioned in a medium, the surface charge distribution of the fixed belt system enables the synthesized electric field to have minimum electrostatic energy, and the expression of the first-class boundary value problem energy integral is as follows:
the electrostatic field problem is therefore equivalent to the functional extremum problem:
similarly, the equivalence functional extreme value problems of the second and third types of edge value problems are as follows:
therein, functional
Quadratic dependence function
And partial derivatives thereof, hence the term
Is a function of
The corresponding variation problem is called a quadratic functional extreme value problem.
Specifically, the second and third types of boundary condition problems are included in the variation among the requirements for the functional to reach the extremum. And the boundary conditions on the interfaces of different media are also included in the requirement that the functional reaches the extreme value and are automatically met. These conditions are therefore referred to as natural boundary conditions. For the first class of boundary conditions, which need to be given as definite conditions in the variational problem, such boundary conditions are called imposed boundary conditions.
In this embodiment, in S4, the method for performing calculation analysis on the unit energy value includes the following steps:
s4.1, performing secondary functional calculation of all unit energies;
s4.2, sequentially carrying out energy integral calculation of the single unit;
s4.3, calculating unit energy integrals which are independent of x, y and z respectively;
and S4.4, simultaneously combining the equations of all the sub-regions.
In this embodiment, in S4.1, performing the quadratic functional calculation on all unit energies may be expressed as a sum of all unit energy functions:
in this embodiment, in S4.2, the expression for performing the single-unit energy integral calculation is as follows:
and the following steps:
thus:
namely:
in this embodiment, in S4.3, the expressions for the integral calculation of the unit energy independent of x, independent of y, and independent of z are:
if it is
Independent of x, then:
wherein:
wherein S isεThe area enclosed by the subdivision area;
in the same way, if
Independently of y, then:
wherein S isεThe area enclosed by the subdivision area;
in the same way, if
Independent of z, then:
wherein S isεThe area enclosed by the subdivision area.
In this embodiment, in S4.4, the expression for simultaneous combination of equations is:
from equation (12) -equation (22), it can be derived:
by combining equations (23) for all sub-regions concurrently, equation (11) can be written as:
wherein
And K is the matrix of variables and coefficients after all the equation sets are connected.
In this embodiment, in S5, the expression of discretizing the functional problem and then performing multivariate quadratic function extremum calculation is:
from the theory of extreme values of functions, there are:
the formula is as follows:
wherein, the formula (27) is a finite element equation, and the equation (27) is solved in sequence to obtain the node of each node
The values may be used to obtain the desired approximate solution.
In this embodiment, in S6, the expression for calculating the electric field strength vector is:
the electric field intensity vector E of each potential can be obtained by combining the potential distribution.
Computer program product embodiment
Referring to fig. 3, a schematic structural diagram of a device for calculating a three-dimensional space electric field based on the finite element method according to the present embodiment is shown, where the device includes a processor, a memory, and a computer program stored in the memory and running on the processor.
The processor comprises one or more than one processing core, the processor is connected with the processor through a bus, the memory is used for storing program instructions, and the method for calculating the three-dimensional space electric field based on the finite element method is realized when the processor executes the program instructions in the memory.
Alternatively, the memory may be implemented by any type or combination of volatile or non-volatile memory devices, such as Static Random Access Memory (SRAM), electrically erasable programmable read-only memory (EEPROM), erasable programmable read-only memory (EPROM), programmable read-only memory (PROM), read-only memory (ROM), magnetic memory, flash memory, magnetic or optical disks.
In addition, the present invention also provides a computer-readable storage medium, in which a computer program is stored, and the computer program, when executed by a processor, implements the steps of the method for calculating a three-dimensional space electric field based on the finite element method.
Optionally, the present invention further provides a computer program product containing instructions, which when run on a computer, causes the computer to perform the steps of the method for calculating a three-dimensional space electric field based on the finite element method according to the above aspects.
It will be understood by those skilled in the art that all or part of the steps for implementing the above embodiments may be implemented by hardware, or may be implemented by hardware related to instructions of a program, which may be stored in a computer-readable storage medium, such as a read-only memory, a magnetic or optical disk, and the like.
The foregoing shows and describes the general principles, essential features, and advantages of the invention. It will be understood by those skilled in the art that the present invention is not limited to the embodiments described above, and the preferred embodiments of the present invention are described in the above embodiments and the description, and are not intended to limit the present invention. The scope of the invention is defined by the appended claims and equivalents thereof.
Claims (10)
1. A three-dimensional space electric field calculation method based on a finite element method is characterized in that: the method comprises the following steps:
s1, selecting COMSOL software based on a finite element method as a principle, and performing three-dimensional modeling and space simulation on the three-dimensional space electric field;
s2, calculating and analyzing the electric field boundary value of the simulated three-dimensional electric field space;
s3, calculating and analyzing the minimum electrostatic energy boundary value problem;
s4, subdividing the simulation model, and calculating and analyzing the unit energy value;
s5, discretizing the functional problem, then performing multivariate quadratic function extremum calculation, and calculating the approximate value required by each node;
and S6, calculating an electric field intensity vector by combining the potential distributions.
2. The finite element method-based method for calculating the electric field in the three-dimensional space according to claim 1, wherein: in S2, the calculation expression of the electric field boundary value is:
in the electrostatic field, the electric field intensity vector and the electric displacement vector meet the loop theorem and the Gaussian theorem, rho is the charge density of the free body, and equation (1) and equation (2) are differential equations used for electrostatic field analysis in the Maxwell equation set;
will be provided with
Substituting formula (2) and considering D ═ E, one can obtain:
in the formula (3), ε is a dielectric constant, and an identity is used
Equation (3) can be expressed as:
the formula (4) is a function of potential
And bulk charge density p, dielectric constant epsilon is constant in a region of uniform medium,
equation (4) can be simplified to:
the expression of equation (5) in the rectangular coordinate system is:
in the region where ρ is 0, there are:
equation (6) is called the poisson equation of the electrostatic field, and equation (7) is called the laplace equation of the electrostatic field, where the laplace equation is a special case of the poisson equation.
3. The finite element method-based method for calculating the electric field in the three-dimensional space according to claim 2, wherein: in S3, the minimum electrostatic energy margin has three types of problems, and the expression is:
the electrostatic field minimum action principle is adopted, a fixed belt system is positioned in a medium, the surface charge distribution of the fixed belt system enables the synthesized electric field to have minimum electrostatic energy, and the expression of the first-class boundary value problem energy integral is as follows:
the electrostatic field problem is therefore equivalent to the functional extremum problem:
similarly, the equivalence functional extreme value problems of the second and third types of edge value problems are as follows:
therein, functional
Quadratic dependence function
And partial derivatives thereof, hence the term
Is a function of
The corresponding variation problem is called a quadratic functional extreme value problem.
4. The finite element method-based method for calculating the electric field in the three-dimensional space according to claim 3, wherein: in S4, the method for performing calculation analysis on the unit energy value includes the following steps:
s4.1, performing secondary functional calculation of all unit energies;
s4.2, sequentially carrying out energy integral calculation of the single unit;
s4.3, calculating unit energy integrals which are independent of x, y and z respectively;
and S4.4, simultaneously combining the equations of all the sub-regions.
5. The finite element method-based method for calculating the electric field in the three-dimensional space according to claim 4, wherein: in S4.1, performing a quadratic functional calculation on all unit energies may be expressed as a sum of all unit energy functions:
6. the finite element method-based method for calculating the electric field in the three-dimensional space according to claim 5, wherein: in S4.2, the expression for performing the single-unit energy integral calculation is as follows:
and the following steps:
thus:
namely:
7. the finite element method-based method for calculating the electric field in the three-dimensional space according to claim 6, wherein: in S4.3, the expressions for the integral calculation of the unit energy independent of x, independent of y, and independent of z are:
if it is
Independent of x, then:
wherein:
wherein S isεThe area enclosed by the subdivision area;
in the same way, if
Independently of y, then:
wherein S isεThe area enclosed by the subdivision area;
in the same way, if
Independent of z, then:
wherein S isεThe area enclosed by the subdivision area.
8. The finite element method-based method for calculating the electric field in the three-dimensional space according to claim 7, wherein: in S4.4, the expression for simultaneous combination of equations is:
from equation (12) -equation (22), it can be derived:
by combining equations (23) for all sub-regions concurrently, equation (11) can be written as:
wherein
And K is the matrix of variables and coefficients after all the equation sets are connected.
9. The finite element method-based method for calculating the electric field in the three-dimensional space according to claim 8, wherein: in S5, the expression of discretizing the functional problem and then performing multivariate quadratic function extremum calculation is:
from the theory of extreme values of functions, there are:
the formula is as follows:
wherein, the formula (27) is a finite element equation, and the equation (27) is solved in sequence to obtain the node of each node
The values may be used to obtain the desired approximate solution.
10. The finite element method-based method for calculating the electric field in the three-dimensional space according to claim 9, wherein: in S6, the expression for calculating the electric field strength vector is:
the electric field intensity vector E of each potential can be obtained by combining the potential distribution.
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| CN114048589A (en) * | 2021-10-19 | 2022-02-15 | 中国南方电网有限责任公司超高压输电公司检修试验中心 | Electric field determining method and device, electronic equipment and storage medium |
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| CN103412989A (en) * | 2013-08-01 | 2013-11-27 | 电子科技大学 | Parameterized reduced model based three-dimensional electromagnetic field simulation method of periodic structure |
| CN104834765A (en) * | 2014-12-31 | 2015-08-12 | 国网电力科学研究院武汉南瑞有限责任公司 | Power cable accessory state evaluation method based on finite element electric field analysis |
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| CN114048589A (en) * | 2021-10-19 | 2022-02-15 | 中国南方电网有限责任公司超高压输电公司检修试验中心 | Electric field determining method and device, electronic equipment and storage medium |
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