CN109408927B - Two-dimensional static magnetic field parallel finite element acceleration method based on black box transmission line model - Google Patents
Two-dimensional static magnetic field parallel finite element acceleration method based on black box transmission line model Download PDFInfo
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Abstract
The invention discloses a finite element solving method of a two-dimensional nonlinear static magnetic field model based on a black box transmission line model, which comprises the following steps: firstly, determining variables to be solved and a solution domain; secondly, establishing an rz rectangular coordinate system; thirdly, listing a control equation and a boundary condition formula in the two-dimensional axisymmetric nonlinear static magnetic field and forming a differential equation set; fourthly, carrying out network division on the solution domain; fifthly, calculating a coefficient matrix of the finite element; sixthly, establishing an equivalent black box circuit model; seventhly, inserting a section of transmission line segment between the black box circuit model and the linear circuit; eighthly, iteration of a transmission line method is carried out; and ninthly, repeating iteration until the iteration result converges to a fixed error, and ending the solution. The method can solve and calculate the solved problem in parallel, and can be used for solving a complex finite element networking model, thereby solving the problems of long solving time and low efficiency caused by solving the finite element nonlinear problem by a Newton iteration method.
Description
Technical Field
The invention belongs to the technical field of electromagnetic field numerical calculation, relates to a finite element solving method of a two-dimensional nonlinear static magnetic field model, and particularly relates to a method for converting a finite element problem of a two-dimensional nonlinear static magnetic field into an equivalent black box circuit problem and realizing parallel accelerated solving of the finite element of the two-dimensional nonlinear static magnetic field by means of a transmission line iteration method.
Background
The finite element method is a numerical calculation method which is most widely applied in the industrial field and is adopted by a plurality of commercial simulation software. However, with the increasing complexity of the solution model and the increasing number of the sub-grid units, the nonlinear finite element solution method using the traditional newton iteration method as the core faces the problem of consuming time for solution, which is directly related to the speed and efficiency of the simulation development of the product.
The core of the finite element problem solution lies in solving a linear equation set, for a nonlinear problem, a new iteration result is used for regenerating a global matrix of a finite element model in each step of a traditional Newton iteration method, the dimension of the global matrix is continuously increased along with the continuous increase of the model sub-network, the time consumed by LU decomposition and the like of each step of the matrix is correspondingly increased, and the total solution time may be increased in a geometric form along with the densification of the sub-network. The transmission line iteration method can avoid repeated LU decomposition process and solving time. However, the conventional transmission line iteration method can only be used in finite element models of first-order triangle elements, and cannot be used in complex networking element models. Therefore, a new iteration method needs to be researched to improve the application range of the transmission line method and the capability of coping with a complex model, so that the problems of long solving time and low efficiency caused by solving the finite element nonlinearity problem by the Newton iteration method are solved.
Disclosure of Invention
The invention aims to provide a finite element solving method of a two-dimensional nonlinear static magnetic field model based on a black box transmission line model, which can solve and calculate the solved problem in parallel and can be used for solving a complex finite element networking model, wherein the complex finite element networking model comprises a triangular unit, a quadrilateral unit and the like, so that the problems of long solving time and low efficiency caused by solving the finite element nonlinear problem by a Newton iteration method are solved.
The purpose of the invention is realized by the following technical scheme:
a finite element solving method of a two-dimensional nonlinear static magnetic field model based on a black box transmission line model comprises the following steps:
firstly, determining variables to be solved and a solution domain, wherein: the variable to be solved is the magnetic potential A of a two-dimensional axisymmetric nonlinear static magnetic field, and the solution domain is the region where the two-dimensional axisymmetric nonlinear static magnetic field is located;
secondly, establishing a rz rectangular coordinate system;
thirdly, listing a control equation and a boundary condition formula in the two-dimensional axisymmetric nonlinear static magnetic field and forming a differential equation set, wherein:
the control equation is:
wherein J is a current density variable, μ ═ r μ, μ is the permeability of a triangular unit, a ═ rA, a is the magnetic potential, r is the abscissa, and z is the ordinate;
the boundary condition formula is as follows:
Γ1:A′＝0；
in the formula, m is a normal vector on a solution boundary;
fourthly, adopting a net separating program to separate the solving domain, dispersing the solving domain into triangular units or quadrilateral units, and in each finite element unit, carrying out variable quantity
Wherein N is the node number of the finite element unit, NjIs a function of the shape in the cell,
the variable value of the corresponding node is the size;
fifthly, calculating a coefficient matrix of the finite element according to the following formula:
wherein the computational expression of each term is as follows:
sixthly, combining the matrix K in the step fiveeAdmittance matrix viewed as a circuit, beRegarding the current source vector as a current source vector, and establishing an equivalent black box circuit model;
seventhly, inserting a section of transmission line segment at any position of a connecting wire between the black box circuit model and the linear circuit;
eighthly, iteration of a transmission line method is carried out:
the voltage signal is incident into a linear network, and the circuit, namely an equation system is solved: (Y)linear+YTL)A′＝b+2ViYTLThroughout the iteration, the matrix (Y)linear+YTL) Keeping unchanged, executing LU decomposition operation of the primary matrix in the 1 st step of iteration, and calculating the reflected voltage V after the solution is completed without executing the subsequent steps againr＝A′-Vi；
And (3) inputting the voltage signal into a nonlinear finite element unit, and solving the circuit, namely a nonlinear equation system:
and (4) solving by adopting a Newton iteration method, wherein the iteration formula of the k step is as follows:
wherein:
solving the equation set in each nonlinear unit is independently put into an independent computing core to be solved, parallel computing is realized, and after the solving is finished, the reflected voltage is calculated again
In the formula, YlinearAn admittance matrix that is a linear circuit incident within the linear network; y isTLAn admittance matrix that is a transmission line circuit incident within the linear network; a' is the voltage of the node to be solved; vrIs the reflected voltage reflected to the non-linear circuit; viIs the incident voltage incident to the linear circuit; keA coefficient matrix which is a nonlinear unit;
an admittance matrix that is reflected back to the transmission line within the nonlinear element;
is the reflected voltage reflected back into the nonlinear cell; j. the design is a squareeIs a Jacobian matrix;
is the Jacobian matrix at the k iteration; f is an intermediate variable in the calculation; a. thee′The value of the magnetic potential for each node in the cell.
And ninthly, repeating the iteration of the step eight until the iteration result converges to a fixed error, and ending the solution.
Compared with the prior art, the invention has the following advantages:
1. and (3) carrying out discrete modeling on the static magnetic field by adopting a finite element method, and enabling the element coefficient matrix of each element in the finite element to be equivalent to a sealed black box circuit model, thereby realizing the equivalent of a finite element equation set to be a nonlinear circuit network problem.
2. And a transmission line segment is inserted between the black box model of the nonlinear unit and the network of the linear unit, and transmission line iteration is realized by utilizing the iteration rule of the transmission line, so that the solution of the static magnetic field problem is realized.
3. In the transmission line iteration process, each nonlinear black box model is independent, and the solution calculation of the nonlinear black box model can adopt independent calculation resources (CPU or GPU core) for calculation, so that parallel acceleration is realized.
4. The iterative solution of finite elements can be carried out by adopting a transmission line method and a black box model, and various complex finite element models such as triangular units and quadrilateral units can be processed.
5. In the iterative solution process, the global matrix Y can be kept unchanged, in the matrix solution process, an LU decomposition method is adopted, LU decomposition is only needed to be carried out in the first calculation step, and because LU decomposition generally occupies about 95% of the time of matrix solution, the method can greatly reduce the iteration time of each step.
6. The nonlinear unit and the linear solving area can be isolated by using a transmission line method, and the parallel acceleration effect is realized.
Drawings
FIG. 1 is a black box circuit model;
FIG. 2 is a black box circuit network after insertion of transmission lines;
FIG. 3 is an equivalent circuit diagram after voltage is incident on the linear network;
FIG. 4 is an equivalent circuit diagram after voltage is reflected back to the nonlinear element;
FIG. 5 is a flow chart of an iterative solution;
FIG. 6 is a block diagram of a contactor of a certain type;
FIG. 7 is a cross-sectional view of a type of contactor;
FIG. 8 is a simplified model of a contactor;
FIG. 9 is a contactor calculation model;
FIG. 10 shows the result of the screening process, (a) triangular units, (b) quadrilateral units;
FIG. 11 is a diagram illustrating a magnetic field distribution in different states obtained by solving using a transmission line method;
FIG. 12 is a single step calculation time acceleration effect.
Detailed Description
The technical solutions of the present invention are further described below with reference to the drawings, but the present invention is not limited thereto, and modifications or equivalent substitutions may be made to the technical solutions of the present invention without departing from the scope of the technical solutions of the present invention.
The first embodiment is as follows: the embodiment provides a finite element solving method of a two-dimensional nonlinear static magnetic field model based on a black box transmission line model, which comprises the following steps:
the method comprises the steps of firstly, determining a variable to be solved and a solving domain, wherein the variable to be solved is a magnetic potential A of a two-dimensional axisymmetric nonlinear static magnetic field, the two-dimensional axisymmetric nonlinear static magnetic field is generated by current in an electrified coil, all elements around the electrified coil are made of ferromagnetic materials, and the solving domain is a region where the two-dimensional axisymmetric nonlinear static magnetic field is located.
And secondly, establishing a rz rectangular coordinate system, wherein r is a horizontal axis and z is a vertical axis.
Thirdly, listing a control equation and a boundary condition formula in the two-dimensional axisymmetric nonlinear static magnetic field and forming a differential equation set, wherein the control equation is as follows:
wherein J is a current density variable, μ ═ r μ, μ is a magnetic permeability of a triangular unit, a ═ rA, a is a magnetic potential, r is an abscissa, z is an ordinate, and boundary conditions are as follows: gamma-shaped1:A′＝0；
And m is a normal vector on the solution boundary.
And fourthly, carrying out network division on the solving area by adopting a network division program, and dispersing the solving area into triangular units or quadrilateral units. In each finite element, the variable
Wherein N is the node number of the finite element unit, NjIs a function of the shape in the cell,
is the magnitude of the variable value of the corresponding node.
And fifthly, calculating a coefficient matrix of the finite element. According to the Galerkin method, carrying out weighted integration on a control equation to obtain an equation:
written in matrix form KeAe′＝beNamely:
wherein the computational expression of each term is as follows:
sixthly, combining the matrix K in the step fiveeAdmittance matrix viewed as a circuit, beRegarded as a current source vector, an equivalent black box circuit model is established, as shown in fig. 1.
And seventhly, the black box model and the linear circuit are connected through a lead, and a transmission line segment is inserted into any position of the lead between the black box model and the linear circuit, as shown in fig. 2.
Eighthly, iteration of a transmission line method is carried out.
The voltage signal is incident on the linear network and, as shown in fig. 3, the circuit, i.e. the system of equations, is solved: (Y)linear+YTL)A′＝b+2ViYTLThroughout the iteration, the matrix (Y)linear+YTL) Keeping unchanged, executing LU decomposition operation of the primary matrix in the 1 st step of iteration, and calculating the reflected voltage V after the solution is completed without executing the subsequent steps againr＝A′-Vi。
The voltage signal is incident on the nonlinear finite element, and as shown in fig. 4, the circuit, i.e. the nonlinear equation system, is solved:
the matrix is solved by adopting a Newton iteration method, and the iteration formula of the k step is as follows:
wherein:
because each nonlinear unit is isolated from each other, the solution of the equation set in each unit is independently put into an independent computing core for solution, and parallel computing is realized. After the solution is completed, the reflected voltage is calculated again
In the formula, YlinearIs the admittance matrix of the linear circuit in fig. 3; y isTLAn admittance matrix for the transmission line circuit within the dashed box of fig. 3; a' is the voltage of the node to be solved; vrIs the reflected voltage reflected to the non-linear circuit; viIs the incident voltage incident to the linear circuit; keA coefficient matrix which is a nonlinear unit;
an admittance matrix for the transmission lines of fig. 4;
is the reflected voltage reflected back into the cell; j. the design is a squareeIs a Jacobian matrix;
is the jacobian matrix at the kth iteration.
And continuously iterating until the iteration result converges to a fixed error, and ending the solution. The solving flow chart is shown in figure 5.
The second embodiment is as follows: the embodiment provides a finite element solving method of a two-dimensional nonlinear static magnetic field model based on a black box transmission line model, which comprises the following concrete implementation steps:
the method comprises the steps of firstly, determining a variable to be solved and a solving domain, wherein the variable to be solved is a magnetic potential A of a two-dimensional axisymmetric nonlinear static magnetic field, the two-dimensional axisymmetric nonlinear static magnetic field is generated by current in an electrified coil, all elements around the electrified coil are made of ferromagnetic materials, and the solving domain is a region where the two-dimensional axisymmetric nonlinear static magnetic field is located. In the present embodiment, the calculation is performed for a certain type of axially symmetric contactor, and the appearance view and the cross-sectional view of the model are shown in fig. 6 and 7, respectively. This is simplified and modeled, the simplified model being shown in fig. 8 and the computational model being shown in fig. 9.
And secondly, establishing a rz rectangular coordinate system, wherein r is a horizontal axis and z is a vertical axis.
Thirdly, listing a control equation and a boundary condition formula in the two-dimensional axisymmetric nonlinear static magnetic field and forming a differential equation set, wherein the control equation is as follows:
wherein J is a current density variable, μ ═ r μ, μ is a magnetic permeability of a triangular unit, a ═ rA, a is a magnetic potential, r is an abscissa, z is an ordinate, and boundary conditions are as follows: gamma-shaped1:A′＝0；
And m is a normal vector on the solution boundary.
Fourthly, adopting a network division program to carry out network division on the solving areaThe solution domain is discretized into triangular units or quadrilateral units, and the result of the division is shown in fig. 10. In each finite element, the variable
Wherein N is the node number of the finite element unit, NjIs a function of the shape in the cell,
is the magnitude of the variable value of the corresponding node.
For the case of a triangular cell,
delta is the area of the triangular unit cell,
for a unit of a quadrilateral shape,
and fifthly, calculating a coefficient matrix of the finite element. According to the Galerkin method, carrying out weighted integration on a control equation to obtain an equation:
written in matrix form KeAe′＝beNamely:
wherein the computational expression of each term is as follows:
sixthly, combining the matrix K in the step fiveeAdmittance matrix viewed as a circuit, beRegarded as a current source vector, an equivalent black box circuit model is established, as shown in fig. 1.
And seventhly, inserting a transmission line segment between the black box model and the linear circuit, as shown in figure 2.
Eighthly, iteration of a transmission line method is carried out.
The voltage signal is incident on the linear network and, as shown in fig. 3, the circuit, i.e. the system of equations, is solved: (Y)linear+YTL)A′＝b+2ViYTLThen, the reflected voltage V is calculatedr＝A′-Vi。
The voltage signal is incident on the nonlinear finite element, and as shown in fig. 4, the circuit, i.e. the nonlinear equation system, is solved:
the matrix is solved by adopting a Newton iteration method, and the iteration formula of the k step is as follows:
wherein:
after the solution is completed, the reflected voltage is calculated again
And continuously iterating until the iteration result converges to a fixed error, and ending the solution. The solving flow chart is shown in figure 5.
The magnetic field distribution conditions in different states are obtained by solving with a transmission line method, and the result is shown in fig. 11.
TABLE 1 different solving models
case 1-5 is a triangular unit sub-net; case 6-10 is a quadrilateral unit sub-net.
TABLE 2 error comparison of N-R iteration and BB-TLM iteration of the present invention
No.1-5 is the error comparison of any five points in the case 5 model of Table 1; no.6-10 are error comparisons of five points arbitrarily taken for the case10 model of Table 1.
Comparing the single step calculation time of the newton iteration method and the transmission line method, as shown in fig. 12, the single step calculation time of the transmission line method is several times faster than that of the newton iteration method, compared with the single step calculation time of the newton iteration method and the transmission line method, which is accelerated by using 1, 4, 8, 16, 20 CPU cores in the middle case10 model in table 1.
Compared with the total calculation time of the newton iteration method and the transmission line method, the acceleration effect of the transmission line method is more obvious than that of the newton iteration method when the CPU cores are increased, in the case10 model in table 1, the CPU cores are accelerated by using 1, 4, 8, 16 and 20 respectively.
Table 3 solving the model in Table 1 by Newton's iteration method and transmission line method, respectively (CPU core 20)
As can be seen from table 3, when the grid division model is increased, the computation time of the transmission line method is faster than that of the newton iteration method.
Claims (1)
1. A finite element solving method of a two-dimensional nonlinear static magnetic field model based on a black box transmission line model is characterized by comprising the following steps:
firstly, determining variables to be solved and a solution domain, wherein: the variable to be solved is the magnetic potential A of a two-dimensional axisymmetric nonlinear static magnetic field, and the solution domain is the region where the two-dimensional axisymmetric nonlinear static magnetic field is located;
secondly, establishing a rz rectangular coordinate system;
thirdly, listing a control equation and a boundary condition formula in the two-dimensional axisymmetric nonlinear static magnetic field and forming a differential equation set, wherein:
the control equation is:
wherein J is a current density variable, μ ═ r μ, μ is the permeability of a triangular unit, a ═ rA, a is the magnetic potential, r is the abscissa, and z is the ordinate;
the boundary condition formula is as follows:
Γ1:A′＝0；
Γ2:
in the formula, m is a normal vector on a solution boundary;
fourthly, adopting a net separating program to separate the solving domain, dispersing the solving domain into triangular units or quadrilateral units, and in each finite element unit, carrying out variable quantity
Wherein N is the node number of the finite element unit, NjIs a function of the shape in the cell,
the variable value of the corresponding node is the size;
fifthly, calculating a coefficient matrix of the finite element according to the following formula:
wherein the computational expression of each term is as follows:
sixthly, combining the matrix K in the step fiveeAdmittance matrix viewed as a circuit, beRegarding the current source vector as a current source vector, and establishing an equivalent black box circuit model;
seventhly, inserting a section of transmission line segment at any position of a connecting wire between the black box circuit model and the linear circuit;
eighthly, iteration of a transmission line method is carried out:
the voltage signal is incident into a linear network, and the circuit, namely an equation system is solved: (Y)linear+YTL)A′＝b+2ViYTLThroughout the iteration, the matrix (Y)linear+YTL) Keeping unchanged, executing LU decomposition operation of the primary matrix in the 1 st step of iteration, and calculating the reflected voltage V after the solution is completed without executing the subsequent steps againr＝A′-Vi；
And (3) inputting the voltage signal into a nonlinear finite element unit, and solving the circuit, namely a nonlinear equation system:
and (4) solving by adopting a Newton iteration method, wherein the iteration formula of the k step is as follows:
solving the equation set in each nonlinear unit is independently put into an independent computing core to be solved, parallel computing is realized, and after the solving is finished, the incident voltage incident to the linear circuit is recalculated
In the formula, YlinearAn admittance matrix that is a linear circuit incident within the linear network; y isTLAn admittance matrix that is a transmission line circuit incident within the linear network; a' is the voltage of the node to be solved; vrIs the reflected voltage reflected to the non-linear circuit; viIs the incident voltage incident to the linear circuit; keA coefficient matrix which is a nonlinear unit;
an admittance matrix that is reflected back to the transmission line within the nonlinear element;
is the reflected voltage reflected back into the nonlinear cell;
is the Jacobian matrix at the k iteration;
and ninthly, repeating the iteration of the step eight until the iteration result converges to a fixed error, and ending the solution.
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