CN109408927A - A kind of two-dimentional magnetostatic field parallel finite element method accelerated method based on black box transmission line model - Google Patents
A kind of two-dimentional magnetostatic field parallel finite element method accelerated method based on black box transmission line model Download PDFInfo
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Abstract
The invention discloses a kind of finite element solving methods of two-dimension non linearity magnetostatic model based on black box transmission line model, and described method includes following steps: one, determining variable to be solved and solve domain；Two, rz rectangular coordinate system is established；Three, governing equation in two-dimensional axial symmetric nonlinear magnetostatic field and boundary condition formula and one differential equation group of composition are listed；Four, subnetting is carried out to solution domain；Five, the coefficient matrix of finite element is calculated；Six, equivalent black box circuit model is established；Seven, one section of transmission line line segment is inserted between black box circuit model and linear circuit；Eight, Transmission line method iteration is carried out；Nine, repeat iteration, until iteration result converges to fixed error, end is solved.The present invention can concurrently carry out solution calculating to required problem, and can be used in complicated finite element meshing model solution, to solve the problems, such as that bring solves time length, low efficiency when Newton iteration method solving finite element nonlinear problem.
Description
Technical field
The invention belongs to Numerical Calculation of Electromagnetic Fields technical field, it is related to a kind of the limited of two-dimension non linearity magnetostatic model
First method for solving, and in particular to a kind of Finite Element Equations by two-dimension non linearity magnetostatic field are converted into black box circuit of equal value and ask
Topic, and realized by transmission line iteration methods and the method solved is accelerated parallel to the finite element of two-dimension non linearity magnetostatic field.
Background technique
FInite Element is most widely used numerical computation method in industrial circle, is used by many commercial simulation softwares.
However, being with traditional Newton iteration method with increasingly complicating for solving model and being increasing for subnetting number of unit
The nonlinear finite element method for solving of core, which is faced with, solves time-consuming serious problem, this emulation for being directly related to product is ground
The speed and efficiency of hair.
The core of the solution of Finite Element Equations is solution system of linear equations, and for nonlinear problem, it is traditional
The each step of Newton iteration method is employed to the global matrix that new iteration result regenerates finite element model, with model point
The time for the consumption such as the continuous increase of net, the dimension of global matrix constantly become larger, and the LU of each step matrix is decomposed can increase accordingly
Greatly, the overall solution time may form geometric increase as the change of subnetting is close.It can be avoided using transmission line iteration methods
Duplicate LU decomposable process solves the time.However, traditional transmission line iteration methods is only used for having for single order triangular element
It limits in meta-model, is not used to complicated subnetting model of element.Therefore, it is necessary to study a kind of new alternative manner, Lai Tigao
The ability of the scope of application of Transmission line method and reply complex model, to solve that Newton iteration method solving finite element is non-linear to ask
Bring solves the problem of long time, low efficiency when topic.
Summary of the invention
The object of the present invention is to provide a kind of the limited of two-dimension non linearity magnetostatic model based on black box transmission line model
First method for solving, this method can concurrently carry out solution calculating to required problem, and can be used in complicated finite element fraction
In pessimistic concurrency control solves, including triangular element and quadrilateral units etc., so that it is non-thread to solve Newton iteration method solving finite element
Property problem when bring the problem of solving time long, low efficiency.
The purpose of the present invention is what is be achieved through the following technical solutions:
A kind of finite element solving method of the two-dimension non linearity magnetostatic model based on black box transmission line model, including such as
Lower step:
One, it determines variable to be solved and solves domain, in which: the variable to be solved is that a two-dimensional axial symmetric is non-
The magnetic potential A of linear magnetostatic field, the domain that solves is the region where two-dimensional axial symmetric nonlinear magnetostatic field；
Two, a rz rectangular coordinate system is established；
Three, governing equation in two-dimensional axial symmetric nonlinear magnetostatic field and boundary condition formula and one differential side of composition are listed
Journey group, in which:
The governing equation are as follows:
In formula, J is current density variable, and μ '=r μ, μ is the magnetic conductivity of triangular unit, and A '=rA, A are magnetic potential, and r is cross
Coordinate, z are ordinate；
The boundary condition formula are as follows:
Γ1: A '=0；
In formula, m is to solve borderline normal vector；
Four, subnetting is carried out to domain is solved using subnetting program, it is discrete for triangular element or quadrangle list that domain will be solved
Member, in each finite element unit, variableIn formula, n is the interstitial content of finite element unit, NjFor list
Shape function in member,For the size of the variate-value of corresponding node；
Five, the coefficient matrix of finite element is calculated according to the following formula:
Wherein, the calculation expression of each single item is as follows:
Six, by the matrix K in step 5eRegard the admittance matrix of circuit, b aseRegard electric current source vector as, establishes equivalent
Black box circuit model；
Seven, one section of transmission line line segment is inserted into the connecting wire any position between black box circuit model and linear circuit；
Eight, Transmission line method iteration is carried out:
Voltage signal is incident in Linear Network, the circuit, i.e. equation group: (Y are solvedlinear+YTL) A '=b+
2ViYTL, in entire iterative process, matrix (Ylinear+YTL) remain unchanged, a matrix is executed in the step 1 of iteration
LU operation splitting, subsequent step without executing again, after the completion of solution, calculates reflected voltage Vr=A '-Vi；
Voltage signal is incident in nonlinear finite element unit, the circuit, i.e. Nonlinear System of Equations are solved:It is solved using Newton iteration method, the iterative formula of kth step is as follows:Wherein:
Solving equations in each non-linear unit are individually put into independent calculating core and solve, and realize parallel
It calculates, after the completion of solution, calculates reflected voltage again
In formula, YlinearAdmittance matrix for the linear circuit being incident in Linear Network；YTLTo be incident on Linear Network
The admittance matrix of interior transmission line circuit；A ' is node voltage to be asked；VrFor the reflected voltage for being reflected into nonlinear circuit；ViFor
It is incident on the incident voltage of linear circuit；KeFor the coefficient matrix of non-linear unit；To be reflected back the biography in non-linear unit
The admittance matrix of defeated line；To be reflected back the reflected voltage in non-linear unit；JeFor Jacobian matrix；Repeatedly for kth time
For when Jacobian matrix；F is the intermediate variable in calculating；Ae′For the magnetic potential value of node each in unit.
Nine, the iteration for repeating step 8, until iteration result converges to fixed error, end is solved.
Compared with the prior art, the present invention has the advantage that
1, discrete modeling is carried out to magnetostatic field using finite element method, by the unit coefficient of each unit in finite element
Matrix is equivalent to the black box circuit model of a sealing, and FEM equations are equivalent to a nonlinear electricity to realize
Road network problem.
2, it is inserted into transmission line line segment between the black-box model of non-linear unit and the network of linear unit, utilizes transmission
The iteration rule of line, realizes transmission line interation, to realize the solution of Magnetostatic Field Problems.
3, in transmission line iterative process, each non-linear black-box model is independent from each other, its solution calculating can
To be calculated using individual computing resource (CPU GPU core), parallel accelerate is realized.
4, it is able to carry out the iterative solution of finite element using Transmission line method and black-box model, is capable of handling Various Complex
Finite element model, such as triangular element and quadrilateral units.
5, during iterative solution, global matrix Y be able to maintain it is constant, in Matrix Solving process, using LU point
Solution, it is only necessary to carry out LU decomposition in the first step of calculating, due to LU decompose it is general occupy Matrix Solving 95% or so when
Between, it will substantially reduce the iteration time of each step in this way.
6, non-linear unit can be isolated using Transmission line method with linear solution region, realizes parallel acceleration effect
Fruit.
Detailed description of the invention
Fig. 1 is black box circuit model；
Fig. 2 is the black box circuit network being inserted into after transmission line；
Fig. 3 is that voltage is incident on the equivalent circuit diagram after Linear Network；
Fig. 4 is that voltage reflection returns the equivalent circuit diagram after non-linear unit；
Fig. 5 is iterative solution flow chart；
Fig. 6 is the structure chart of certain model contactor；
Fig. 7 is the sectional view of certain model contactor；
Fig. 8 is contactor simplified model；
Fig. 9 is contactor computation model；
Figure 10 is subnetting as a result, (a) triangular element, (b) quadrilateral units；
Figure 11 is the Distribution of Magnetic Field situation for solving to obtain using Transmission line method under different conditions；
Figure 12 is that single step calculates time acceleration effect.
Specific embodiment
Further description of the technical solution of the present invention with reference to the accompanying drawing, and however, it is not limited to this, all to this
Inventive technique scheme is modified or replaced equivalently, and without departing from the range of technical solution of the present invention, should all be covered in this hair
In bright protection scope.
Specific embodiment 1: present embodiments provide for a kind of two-dimension non linearity based on black box transmission line model is quiet
The finite element solving method of magnetic field model, described method includes following steps:
One, it determines variable to be solved and solves domain, variable to be solved is a two-dimensional axial symmetric nonlinear magnetostatic field
Magnetic potential A, two-dimensional axial symmetric nonlinear magnetostatic field generates by the electric current in hot-wire coil, and each element around hot-wire coil is equal
For ferromagnetic material, solving domain is the region where two-dimensional axial symmetric nonlinear magnetostatic field.
Two, a rz rectangular coordinate system is established, r is horizontal axis, and z is the longitudinal axis.
Three, governing equation in two-dimensional axial symmetric nonlinear magnetostatic field and boundary condition formula and one differential side of composition are listed
Journey group, governing equation are as follows:Wherein, J is current density variable, and μ '=r μ, μ is
The magnetic conductivity of triangular unit, A '=rA, A are magnetic potential, and r is abscissa, and z is ordinate, boundary condition are as follows: Γ1: A '=0；M is to solve borderline normal vector.
Four, subnetting is carried out to domain using subnetting program, it is discrete for triangular element or quadrangle that domain will be solved
Unit.In each finite element unit, variableWherein, n is the interstitial content of finite element unit, NjFor
Shape function in unit,For the size of the variate-value of corresponding node.
Five, the coefficient matrix of finite element is calculated.According to Galerkin method, integral, the side of obtaining are weighted to governing equation
Journey:
Write as matrix form KeAe′=be, i.e., are as follows:
Wherein, the calculation expression of each single item is as follows:
Six, by the matrix K in step 5eRegard the admittance matrix of circuit, b aseRegard electric current source vector as, establishes equivalent
Black box circuit model, as shown in Figure 1.
It seven, is to be connected between black-box model and linear circuit by conducting wire, between black-box model and linear circuit
These conducting wires any position be inserted into one section of transmission line line segment, as shown in Fig. 2.
Eight, Transmission line method iteration is carried out.
Voltage signal is incident in Linear Network, as shown in figure 3, solving the circuit, i.e. equation group: (Ylinear+YTL)A′
=b+2ViYTL, in entire iterative process, matrix (Ylinear+YTL) remain unchanged, first moment is executed in the step 1 of iteration
The LU operation splitting of battle array, subsequent step without executing again, after the completion of solution, calculates reflected voltage Vr=A '-Vi。
Voltage signal is incident in nonlinear finite element unit, as shown in figure 4, solving the circuit, i.e. nonlinear equation
Group:The matrix is solved using Newton iteration method, and the iterative formula of kth step is as follows:Wherein:
Since each non-linear unit is isolation between each other, the solving equations in each unit are individually put
It is solved into independent calculating core, realizes parallel computation.After the completion of solution, reflected voltage is calculated again
In formula, YlinearFor the admittance matrix of linear circuit in Fig. 3；YTLIt is led for transmission line circuit in Fig. 3 dotted line frame
Receive matrix；A ' is node voltage to be asked；VrFor the reflected voltage for being reflected into nonlinear circuit；ViTo be incident on entering for linear circuit
Radio pressure；KeFor the coefficient matrix of non-linear unit；For the admittance matrix of transmission line in Fig. 4；To be reflected back in unit
Reflected voltage；JeFor Jacobian matrix；Jacobian matrix when iteration secondary for kth.
It is constantly iterated, until iteration result converges to fixed error, end is solved.Solve flow chart such as Fig. 5.
Specific embodiment 2: present embodiments provide for a kind of two-dimension non linearity based on black box transmission line model is quiet
The finite element solving method of magnetic field model, the method specific implementation step are as follows:
One, it determines variable to be solved and solves domain, variable to be solved is a two-dimensional axial symmetric nonlinear magnetostatic field
Magnetic potential A, two-dimensional axial symmetric nonlinear magnetostatic field generates by the electric current in hot-wire coil, and each element around hot-wire coil is equal
For ferromagnetic material, solving domain is the region where two-dimensional axial symmetric nonlinear magnetostatic field.Present embodiment is directed to certain model axis pair
Structure contactor is claimed to be calculated, the outside drawing of the model is as shown in fig. 6, sectional view is as shown in Figure 7.It is simplified,
And model, simplified model is as shown in figure 8, computation model is as shown in Figure 9.
Two, a rz rectangular coordinate system is established, r is horizontal axis, and z is the longitudinal axis.
Three, governing equation in two-dimensional axial symmetric nonlinear magnetostatic field and boundary condition formula and one differential side of composition are listed
Journey group, governing equation are as follows:Wherein, J is current density variable, and μ '=r μ, μ is
The magnetic conductivity of triangular unit, A '=rA, A are magnetic potential, and r is abscissa, and z is ordinate, boundary condition are as follows: Γ1: A '=0；M is to solve borderline normal vector.
Four, subnetting is carried out to domain using subnetting program, it is discrete for triangular element or quadrangle that domain will be solved
Unit, the results are shown in Figure 10 for subnetting.In each finite element unit, variableWherein, n is finite element
The interstitial content of unit, NjFor the shape function in unit,For the size of the variate-value of corresponding node.
For triangular element,Δ is the area of triangular element,
For quadrilateral units,
Five, the coefficient matrix of finite element is calculated.According to Galerkin method, integral, the side of obtaining are weighted to governing equation
Journey:
Write as matrix form KeAe′=be, i.e., are as follows:
Wherein, the calculation expression of each single item is as follows:
Six, by the matrix K in step 5eRegard the admittance matrix of circuit, b aseRegard electric current source vector as, establishes equivalent
Black box circuit model, as shown in Figure 1.
Seven, one section of transmission line line segment is inserted between black-box model and linear circuit, as shown in Figure 2.
Eight, Transmission line method iteration is carried out.
Voltage signal is incident in Linear Network, as shown in figure 3, solving the circuit, i.e. equation group: (Ylinear+YTL)A′
=b+2ViYTL, then, calculate reflected voltage Vr=A '-Vi。
Voltage signal is incident in nonlinear finite element unit, as shown in figure 4, solving the circuit, i.e. nonlinear equation
Group:The matrix is solved using Newton iteration method, and the iterative formula of kth step is as follows:Wherein:
After the completion of solution, reflected voltage is calculated again
It is constantly iterated, until iteration result converges to fixed error, end is solved.Solve flow chart such as Fig. 5.
Solve to obtain the Distribution of Magnetic Field situation under different conditions using Transmission line method, as a result as shown in Figure 11.
The different solving model of table 1
Case 1-5 is triangular element subnetting；Case 6-10 is quadrilateral units subnetting.
The error of 2 N-R iterative method of table and BB-TLM iterative method of the present invention compares
No.1-5 is to appoint that the error of five points is taken to compare in 5 model of table 1case；No.6-10 is 1 case of table, 10 mould
Type, which is appointed, takes the error of five points to compare.
Case10 model in table 1 is accelerated using 1,4,8,16,20 core cpu respectively, compares Newton iteration
The single step of method and Transmission line method calculates the time, and as seen from Figure 12, it is faster than Newton iteration method several that the single step of Transmission line method calculates the time
Times.
Case10 model in table 1 is accelerated using 1,4,8,16,20 core cpu respectively, compares Newton iteration
The total evaluation time of method and Transmission line method, when core cpu increases, the acceleration effect of Transmission line method is more than Newton iteration method
Obviously.
Newton iteration method is respectively adopted for the model in table 1 for table 3 and Transmission line method solves (CPU core is 20)
As shown in Table 3, when subnetting model increases, the calculating time of Transmission line method is faster than Newton iteration method.
Claims (1)
1. a kind of finite element solving method of the two-dimension non linearity magnetostatic model based on black box transmission line model, it is characterised in that
Described method includes following steps:
One, it determines variable to be solved and solves domain, in which: the variable to be solved is a two-dimensional axial symmetric nonlinear static
The magnetic potential A in magnetic field, the domain that solves is the region where two-dimensional axial symmetric nonlinear magnetostatic field；
Two, a rz rectangular coordinate system is established；
Three, governing equation in two-dimensional axial symmetric nonlinear magnetostatic field and boundary condition formula and one differential equation group of composition are listed,
Wherein:
The governing equation are as follows:
In formula, J is current density variable, and μ '=r μ, μ is the magnetic conductivity of triangular unit, and A '=rA, A are magnetic potential, and r is abscissa,
Z is ordinate；
The boundary condition formula are as follows:
Γ1: A '=0；
In formula, m is to solve borderline normal vector；
Four, subnetting is carried out to domain is solved using subnetting program, it is discrete for triangular element or quadrilateral units to solve domain,
In each finite element unit, variableIn formula, n is the interstitial content of finite element unit, NjIn unit
Shape function,For the size of the variate-value of corresponding node；
Five, the coefficient matrix of finite element is calculated according to the following formula:
Wherein, the calculation expression of each single item is as follows:
Six, by the matrix K in step 5eRegard the admittance matrix of circuit, b aseRegard electric current source vector as, establishes equivalent black box
Circuit model；
Seven, one section of transmission line line segment is inserted into the connecting wire any position between black box circuit model and linear circuit；
Eight, Transmission line method iteration is carried out:
Voltage signal is incident in Linear Network, the circuit, i.e. equation group: (Y are solvedlinear+YTL) A '=b+2ViYTL, whole
In a iterative process, matrix (Ylinear+YTL) remain unchanged, the LU operation splitting of a matrix is executed in the step 1 of iteration,
Subsequent step without executing again, after the completion of solution, calculates reflected voltage Vr=A '-Vi；
Voltage signal is incident in nonlinear finite element unit, the circuit, i.e. Nonlinear System of Equations are solved:It is solved using Newton iteration method, the iterative formula of kth step is as follows:Solving equations in each non-linear unit are individually put into independent meter
It calculates and is solved in core, realize parallel computation, after the completion of solution, calculate reflected voltage again
In formula, YlinearAdmittance matrix for the linear circuit being incident in Linear Network；YTLTo be incident on the biography in Linear Network
The admittance matrix of defeated line circuit；A ' is node voltage to be asked；VrFor the reflected voltage for being reflected into nonlinear circuit；ViTo be incident on
The incident voltage of linear circuit；KeFor the coefficient matrix of non-linear unit；For the transmission line that is reflected back in non-linear unit
Admittance matrix；To be reflected back the reflected voltage in non-linear unit；Jacobian matrix when iteration secondary for kth；
Nine, the iteration for repeating step 8, until iteration result converges to fixed error, end is solved.
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