CN106295053A - A kind of transient electromagnetic temperature field based on adaptive time-step coupling calculation - Google Patents

Abstract

The present invention proposes a kind of transient electromagnetic temperature field based on adaptive time-step coupling calculation.Use exponential smoothing prediction electromagnetism temperature field coupling time node.And between two coupling time nodes, by predictor corrector method and response characteristic value computational EM waves and temperature field optimized discrete step-length.With tradition unique step coupling process contrast, electromagnetism, temperature field all use optimal discrete steps, it is to avoid the calculating of temperature field overfrequency, reduce the calculating time.Finally, as a example by the copper lead ring that is energized, use adaptive step coupling to calculate copper conductor temperature rise in 0.1s under alternating current, reduce 20% than traditional unique step coupling process time, it was demonstrated that the effectiveness of the method.

Description

A kind of transient electromagnetic based on adaptive time-step-temperature field coupling calculation

Technical field

A kind of transient electromagnetic based on adaptive time-step-temperature field coupling calculation of the present invention, relates to electromagnetism temperature Degree calculating field.

Background technology

In the equipment such as coil discharger, relay, motor, producing too high temperature rise during operation will affect coil and connects Head electric conductivity, the insulating properties of material, even can produce destructive thermal expansion, produces for equipment runnability and safety Certain impact.In the problems referred to above, the big electric current of transition makes the temperature of conductor increase sharply in the short period of time, thus Not only consider that electromagnetic field, to temperature profile effect, also needs to consider that variations in temperature is to electromagnetic field materials conductive performance in calculating process Impact, needs for this to set up transient electromagnetic-temperature field order strong-coupling model.Current employing finite element model for solving transient electromagnetic- During the INDIRECT COUPLING of temperature field, electromagnetic field and temperature field often use identical time step.But in solution procedure, temperature field Respond relatively electromagnetic field slow, calculate according to identical time step so that solution of Temperature overfrequency, cause the increasing of the time of solving Add.

During for two physical field transient state INDIRECT COUPLING, the problem that physical field response time is different.There is method according to existing soft Part, proposes code coupling conception, each physical field is used different time discrete strategies, and carries on the Coupling point of time The transmission of lotus.Also having method for when fluid-wall interaction, convection cell region uses different time steps with solid area DFMT-SCSS algorithm, chooses the multiple that solid domain time step is fluid domain and calculates.But said method exists When calculating two physic field coupling, simply there is simple multiple proportion in two physical field time steps, and physical field all uses constant , there are two physical fields and do not obtain the situation of optimized discrete strategy in step size computation.Adaptive step coupling conception is proposed, respectively for this Physical field uses adaptive step algorithm to obtain optimized discrete time parameter method, and coupling automatically on the coupling time node of prediction Close.Each physical field can be made to obtain optimal time discrete strategy, it also avoid the calculating of the slower physical field overfrequency of response.

Summary of the invention

For solving above-mentioned technical problem, for Transient Electromagnetic-temperature field INDIRECT COUPLING problem, the present invention propose a kind of based on The transient electromagnetic of adaptive time-step-temperature field coupling calculation.At T_nIn the moment, put down as it is shown in figure 1, first pass through index Sliding method predicts next coupling time point T_n+1.Then between two coupling time nodes, by prediction-correction methods and response characteristic value Computational EM waves and temperature field optimized discrete step-length.Obtain T_nTo T_n+1In time period, electromagnetic field and temperature field Best Times are discrete Step-length is △ t_n ^EWith △ t_n ^T.Electromagnetic field, temperature field are respectively adopted △ t_n ^E、△t_n ^TCalculate to coupling time node.Make each physical field On two coupling time nodes, calculate with optimal time step, it is to avoid the calculating of the physical field overfrequency that response is slower.

The technical solution adopted in the present invention is:

A kind of transient electromagnetic based on adaptive time-step-temperature field coupling calculation, comprises the following steps:

1), initialize.Initially set up electromagnetic field and the temperature computation FEM (finite element) model analyzing object, and given electromagnetic field meter The material parameters such as the specific heat capacity of pcrmeability, resistivity and Temperature calculating in calculation, thermal conductivity.Finally apply initial condition, limit Boundary and load；

2), coupling time point determines.Use temperature field to trigger heat and judge electromagnetism temperature field coupling time node.First Electromagnetism-temperature field uses less step-length to couple three times, is used for obtaining temperature field and triggers heat Q_prePrediction data.Then use Heat is triggered in exponential smoothing predicted temperature field, judges coupling time node by triggering heat；

3), electromagnetic field adaptive step calculates.Determine that electromagnetic field load is by load discretization error and response characteristic value Good discrete step-length △ t_n ^E.Start electromagnetic field, use optimized discrete step size computation.Touch when electromagnetic field accumulation heat reaches temperature field During caloric value, suspend Electromagnetic Calculation；

4), temperature field adaptive step calculates.With 3) in, determine electromagnetic field by load discretization error and response characteristic value The step-length △ t of load optimized discrete_n ^T.Electromagnetic Calculation evenly heat power is added to temperature field as load, works as Temperature calculating When time is Tong Bu with electromagnetic field, suspend Temperature calculating.Update electromagnetic field node temperature, and carry out the temperature field of future time step Trigger heat Calculation.Iterate calculating to the final time.

A kind of transient electromagnetic based on adaptive time-step-temperature field coupling calculation of the present invention, advantage is:

1), due to order coupling time, in the single time step of computational EM waves, do not consider the change in temperature field, because of And there is the unbalanced error of accumulation.Use temperature field to trigger heat and judge coupling time node, can effectively control non-flat Weighing apparatus error.

2), compared with tradition unique step coupling process, it is to avoid owing to temperature field response time is than electromagnetism head, and use Unified calculation time step causes solution of Temperature number of times too much to increase the problem of calculating time, reduces the calculating time.

Accompanying drawing explanation

Fig. 1 is electromagnetism-temperature field self adaptation coupling schematic diagram.

Fig. 2 is electromagnetism-temperature field adaptive step coupling flow chart.

Fig. 3 is load discretization error figure.

Fig. 4 (a) is copper guide ring structure schematic diagram；

Fig. 4 (b) is FEM (finite element) model figure.

Fig. 5 (a) unique step temperature computation temperature profile；

Fig. 5 (b) adaptive step temperature profile.

Fig. 6 (a) is No. 1 unit result of calculation figure；

Fig. 6 (b) is No. 140 unit result of calculation figures.

Detailed description of the invention

A kind of transient electromagnetic based on adaptive time-step-temperature field coupling calculation, comprises the following steps:

1), initialize.Initially set up electromagnetic field and the temperature computation FEM (finite element) model analyzing object, and given electromagnetic field meter The material parameters such as the specific heat capacity of pcrmeability, resistivity and Temperature calculating in calculation, thermal conductivity.Finally apply corresponding initial strip Part, border and load；

2), coupling time point determines.Use temperature field to trigger heat and judge electromagnetism temperature field coupling time node.First Electromagnetism-temperature field uses less step-length to couple three times, is used for obtaining temperature field and triggers heat Q_prePrediction data.Then use Heat is triggered in exponential smoothing predicted temperature field, judges coupling time node by triggering heat；

3), electromagnetic field adaptive step calculates.Determine that electromagnetic field load is by load discretization error and response characteristic value Good discrete step-length △ t_n ^E.Start electromagnetic field, use optimized discrete step size computation.Touch when electromagnetic field accumulation heat reaches temperature field During caloric value, suspend Electromagnetic Calculation；

4), temperature field adaptive step calculates.With 3) in, determine electromagnetic field by load discretization error and response characteristic value The step-length △ t of load optimized discrete_n ^T.Electromagnetic Calculation evenly heat power is added to temperature field as load, works as Temperature calculating When time is Tong Bu with electromagnetic field, suspend Temperature calculating.Update electromagnetic field node temperature, and carry out the temperature field of future time step Trigger heat Calculation.Iterate calculating to the final time.

Specifically comprise the following steps that

Step 1): set up Electromagnetic Calculation FEM (finite element) model, and given pcrmeability, resistivity materials parameter, imposed load. Being ignored displacement current by like steady condition, electromagnetic fiele equation can be written as the vector magnetic potential equation of following form:

In formula: Ω₁For conductive region, Ω₂For non-conductor region.A is vector magnetic potential (Wb/m)；V is current potential (V)；σ is electricity Conductance (S/m)；μ is pcrmeability (H/m)；J_sFor source electric current density (A/m²)。

Use Galerkin method that above formula write as finite element scheme:

In formula:R is electromagnetic field damping matrix；S is electromagnetic field coefficient matrix；J is magnetic field load matrix.

Step 2): set up field, temperature field and calculate FEM (finite element) model, and given specific heat capacity, conductivity material parameter.Only examining Considering under conduction of heat and concurrent condition, temperature field Heat Conduction Differential Equations can be written as form:

In formula: ρ is density (Kg/m), C_pFor specific heat capacity (J/ (Kg.K)), Q is heating power W.

Initial condition and boundary condition be:

In formula: (x, y z) represent initial temperature distribution to G；(x, y z) represent steady temperature boundary condition to F；Γ_qRepresent and dissipate Heat boundary condition, q is border heating power, and h is the boundary convection coefficient of heat transfer.

Using Galerkin method by formula (3), it is as follows that (4) form finite element scheme:

In formula: C is temperature field specific heat matrix, K is temperature field heat conduction matrix, and P is loading matrix.

Step 3): use temperature field to trigger heat and judge electromagnetism temperature field coupling time node.Electromagnetism-temperature field uses Less step-length couples three times, is used for obtaining temperature field and triggers heat Q_prePrediction data.

Step 4): allow changes in material maximum temperature during obtaining Electromagnetic Calculation.Linear material thermal power P and electric current Density is J relation:

P_n=∫_VJ_n ²ε(T)dV (6)

ε=aT+ ε₀ (7)

In formula: V is unit volume, P_nFor t_nMoment heating power.J_nFor T_nMoment electric current density, ε is resistivity, and a is electricity Resistance varies with temperature rate, ε₀Resistivity when being 0 DEG C.

When input thermal power is P_n, variations in temperature cause power calculation error to meet:

|P_n(T_n)-P_n(T_n+△T)|≤γP_n(T_n) (8)

When formula (8) takes equal sign, T can be drawn_nMoment temperature allows change maximum △ T_max。

Step 5): heat Calculation is triggered in temperature field.According to T_nMoment maximum temperature change △ T_max.By T_n, T_n-1, T_n-2Moment Input power and variations in temperature, it was predicted that T_n+1Heat is triggered in temperature field

First input heat of each moment and variations in temperature are calculated, such as table 1.

Table 1 each moment heat, temperature

Calculate each moment temperature again with heat gradient, such as table 2.

Table 2 moment rate of temperature change

Finally calculate temperature field and trigger heat Calculation.Use EXSMOOTH prediction t_n+1Moment change of temperature field rate K_n+1:

K_n+1=α K_n+(1-α)K_n-1+(1-α)²K_n-2 (9)

Heat is triggered in temperature field:

Q^pre=△ T_maxK_n+1 (10)

Step 6): start electromagnetic field, determine electromagnetic field time step according to load discretization errort∈(t_n-1,t_n) Time, when loading matrix P uses linear interpolation, equivalent load uses slope to load, as shown in Figure 3.By the discrete generation of load Error is such as formula (13).

Formula (13) is used trapezoid formula integration, obtains discretization error and approximate such as formula (14).

According to formula (14), load discretization error is approximately proportional to (△ t)², next step size computation can be divided into following two Step:

(a) step-ahead prediction.Error is calculated, it was predicted that the (n+1)th step-length △ t according to the n-th step¹ _n+1。

In formula:For safety coefficient,e_toleranceFor allowing maximum error；It it is generation in the n-th time step Discretization error.

B () step-length corrects.Judge as the (n+1)th time step △ t¹ _n+1Whether produced error meets e^k+1<e_tolerance.As It is unsatisfactory for using (8) to be modified iterative computation, until meeting e^k<e_tolerance。

In formula:For safety coefficient,

Step 7): determine electromagnetic field time step according to response characteristic valueHughes proposes according to response characteristic λ_rValue Determine calculating step-length △ t stabilization time_n+1Method^[16], define △ t_n+1λ is concussion restrictive condition.As △ t_n+1λ > > 1 time system It is in concussion state.For ensureing that the desirable maximum step-length of computational stability need to meet:

In formula: f < 1, f is the coefficient of stability；λ_rFor response characteristic value；△u_nFor t_n-1To t_nThe change of time period field amount u.

Step 8): electromagnetic field is at t_nMoment discrete time step is:

Step 9): coupling time judges.Work as T_n+1Moment electromagnetic field accumulation heat meets following condition for the moment, T_n+1For electricity Magnetic-temperature field coupling time point, start-up temperature field calculates.

(a) unit maximum heat change reach step 5) in unit prediction heat threshold time automatic start-up temperature field calculate:

(b) overall heat change reach step 5) in unit prediction heat threshold time automatic start-up temperature field calculate:

Step 10): start-up temperature field calculates, and calculates temperature field adaptive time-stepTemperature field auto-adaptive time step Long calculate same step 6), step 7), step 8)；

Step 11): when the Temperature calculating time arrive step 9) in coupling time T_n+1Time, stop Temperature calculating, update Electromagnetic field node temperature.

Step 12): iterate step 4)～step 10), until calculating total time T_total。

Embodiment:

As a example by energising copper lead ring transient temperature rise:

This section illustrates adaptive time-step application in Coupled Electromagnetic-Thermal, this mould as a example by electrolytic copper lead ring heat is analyzed Type is widely present in the device such as relay, coil transmissions.By a long 10mm, thick 2mm copper lead ring is placed in air, such as Fig. 4 (a) Shown in), the upper and lower and surfaces externally and internally convection transfer rate of copper lead ring is 5W/m².Copper lead ring is applied electric current i=10⁴sin(50 π t), persistent period 0.1s, analyzes the variations in temperature of copper ring.

Setting up axisymmetric model, as shown in Fig. 5 (b), Electromagnetic Calculation region comprises copper ring, air section, uses triangle Grid division, totally 988 nodes, 2061 unit.Temperature calculating region is copper ring region, is also adopted by tessellation net Lattice, totally 121 nodes, 200 unit.

When in formula (15), electromagnetic field load discretization error is taken as e_tolerance=0.5%, formula (8), formula (21) take γ= 1%, β=1.Use 1ms coupling in first three time step of electromagnetism-temperature field, use adaptive time-step coupling subsequently.Finally Adaptive time-step result of calculation is contrasted with fixed step size result of calculation, as shown in Fig. 5 (a), Fig. 5 (b), respectively copper lead ring When applying exchange load 0.1s, unique step and adaptive time-step is used to calculate the Temperature Distribution cloud atlas of gained.

From Fig. 6 (a), Fig. 6 (b), use copper ring Temperature Distribution and the fixed step size temperature of adaptive time-step gained It is distributed basically identical.Wherein No. 1 cell temperature is the highest and relative error minimum 0.35%, and No. 140 cell temperatures are minimum and phase Error is to the maximum 0.53%.Choose No. 1 unit and No. 140 unit for this, analyze its temperature and mistake in whole heating process The Changing Pattern of difference.

No. 1 and No. 140 in 0.1s adaptive time-step and fixed step size calculate gained cell temperature to such as Fig. 6 (a), Shown in Fig. 6 (b), copper guide block temperature approximation rose with 0.02s for the cycle.As a example by 0.06s～0.08s, in a and c of region, electricity Stream amplitude is less, and temperature field rises relatively slow, and corresponding Temperature calculating step-length is bigger.And region b, current amplitude is relatively big, in temperature Rising very fast, it is less that corresponding temperature field calculates time step, and the effectiveness of electromagnetism-temperature field auto step extent coupling algorithm is described.? During whole calculating, auto step extent calculates with fixed step size calculating error within 0.7%, understands the accuracy of the method.

Electromagnetism-temperature field adaptive time-step couples calculated performance contrast as shown in table 3 with fixed step size.In 0.1s, Employing fixed step size couples, and when access time, step-length was 1ms, electromagnetic field and temperature field are both needed to calculate 100 times, and the overall calculation time is 235s.And when using adaptive step to solve, Electromagnetic Calculation 93 times, Temperature calculating 29 times, the calculating time is 184s, calculates Time reduces 21%.

Table 3 calculated performance contrasts

Claims (2)

1. transient electromagnetic based on adaptive time-step-temperature field coupling calculation, it is characterised in that include following Step:

1), initialize: initially set up in electromagnetic field and the temperature computation FEM (finite element) model, and given Electromagnetic Calculation of analyzing object The specific heat capacity of pcrmeability, resistivity and Temperature calculating, the material parameter such as thermal conductivity, finally apply initial condition, border and Load；

2), coupling time point determines: triggering heat in employing temperature field is to judge electromagnetism temperature field coupling time node, first electric Magnetic-temperature field uses less step-length to couple three times, is used for obtaining temperature field and triggers heat Q_prePrediction data, then use refer to Heat is triggered in number smoothing techniques predicted temperature field, judges coupling time node by triggering heat；

3), electromagnetic field adaptive step calculate: by load discretization error and response characteristic value determine electromagnetic field load most preferably from The step-length △ t dissipated_n ^E, start electromagnetic field, use optimized discrete step size computation, trigger heat when electromagnetic field accumulation heat reaches temperature field During amount, suspend Electromagnetic Calculation；

4), temperature field adaptive step calculates: with 3) in, determine electromagnetic field load by load discretization error and response characteristic value The step-length △ t of optimized discrete_n ^T, Electromagnetic Calculation evenly heat power is added to temperature field as load, when the Temperature calculating time Time Tong Bu with electromagnetic field, suspend Temperature calculating, update electromagnetic field node temperature, and the temperature field carrying out future time step is triggered Heat Calculation, the calculating that iterates is to the final time.

A kind of transient electromagnetic based on adaptive time-step-temperature field coupling calculation, its It is characterised by comprising the following steps:

Step 1): setting up Electromagnetic Calculation FEM (finite element) model, and given pcrmeability, resistivity materials parameter, imposed load, by seemingly Steady condition ignores displacement current, and electromagnetic fiele equation can be written as the vector magnetic potential equation of following form:

$\begin{array}{c}\left\{\begin{array}{c}\&dtri;\&times;(\frac{1}{\mathrm{\&mu;}}\&dtri;\&times;A)-\&dtri;(\frac{1}{\mathrm{\&mu;}}\&dtri;\&CenterDot;A)+\mathrm{\&sigma;}\frac{\&part;A}{\&part;t}+\\ \mathrm{\&sigma;}\&dtri;V-v\&times;\mathrm{\&sigma;}\&dtri;\&times;A=J_s\\ \begin{array}{cc}\&dtri;\&CenterDot;(-\mathrm{\&sigma;}\frac{\&part;A}{\&part;t}-\mathrm{\&sigma;}\&dtri;V+v\&times;\mathrm{\&sigma;}\&dtri;\&times;A)=0& {\mathrm{in\&Omega;}}_1\end{array}\end{array}\right.\\ \begin{array}{cc}\&dtri;\&times;(\frac{1}{\mathrm{\&mu;}}\&dtri;\&times;A)-\&dtri;(\frac{1}{\mathrm{\&mu;}}\&dtri;\&CenterDot;A)=0& {\mathrm{in\&Omega;}}_2\end{array}\end{array}---\left(1\right)$

In formula: Ω₁For conductive region, Ω₂For non-conductor region, A is vector magnetic potential (Wb/m)；V is current potential (V)；σ is electrical conductivity (S/m)；μ is pcrmeability (H/m)；J_sFor source electric current density (A/m²),

Use Galerkin method that above formula write as finite element scheme:

$R\stackrel{\&CenterDot;}{U}+SU=J---\left(2\right)$

In formula:R is electromagnetic field damping matrix；S is electromagnetic field coefficient matrix；J is magnetic field load matrix；

Step 2): set up field, temperature field and calculate FEM (finite element) model, and given specific heat capacity, conductivity material parameter, only consider heat Under conduction and concurrent condition, temperature field Heat Conduction Differential Equations can be written as form:

${\mathrm{\&rho;C}}_p\frac{\&part;T}{\&part;t}=\&dtri;\&CenterDot;(k\&dtri;T)+Q---\left(3\right)$

In formula: ρ is density (Kg/m), C_pFor specific heat capacity (J/ (Kg.K)), Q is heating power W,

Initial condition and boundary condition be:

$\left\{\begin{array}{ccc}T=F(x,y,z)& on& {\mathrm{\&Gamma;}}_T\\ k\frac{\&part;T}{\&part;n}+q+h(T-T_a)=0& on& {\mathrm{\&Gamma;}}_q\\ T=G(x,y,z)& on& t=0\end{array}\right.---\left(4\right)$

In formula: (x, y z) represent initial temperature distribution to G；(x, y z) represent steady temperature boundary condition to F；Γ_qRepresent heat dissipation capacity limit Boundary's condition, q is border heating power, and h is the boundary convection coefficient of heat transfer,

Using Galerkin method by formula (3), it is as follows that (4) form finite element scheme:

$C\stackrel{\&CenterDot;}{T}+KT=P---\left(5\right)$

In formula: C is temperature field specific heat matrix, K is temperature field heat conduction matrix, and P is loading matrix；

Step 3): use temperature field to trigger heat and judge that electromagnetism temperature field coupling time node, electromagnetism-temperature field use less Step-length couples three times, is used for obtaining temperature field and triggers heat Q_prePrediction data；

Step 4): allow changes in material maximum temperature, linear material thermal power P and electric current density during obtaining Electromagnetic Calculation For J relation:

P_n=∫_VJ_n ²ε(T)dV (6)

ε=aT+ ε₀ (7)

In formula: V is unit volume, P_nFor t_nMoment heating power, J_nFor T_nMoment electric current density, ε is resistivity, a be resistance with Rate of temperature change, ε₀Resistivity when being 0 DEG C；

When input thermal power is P_n, variations in temperature cause power calculation error to meet:

|P_n(T_n)-P_n(T_n+△T)|≤γP_n(T_n) (8)

When formula (8) takes equal sign, T can be drawn_nMoment temperature allows change maximum △ T_max。

Step 5): heat Calculation is triggered in temperature field, according to T_nMoment maximum temperature change △ T_max.By T_n, T_n-1, T_n-2Moment inputs Power and variations in temperature, it was predicted that T_n+1Heat is triggered in temperature field

First input heat of each moment and variations in temperature are calculated, such as table 1:

Table 1 each moment heat, temperature

Calculate each moment temperature again with heat gradient, such as table 2:

Table 2 moment rate of temperature change

Finally calculate temperature field and trigger heat Calculation, use EXSMOOTH prediction t_n+1Moment change of temperature field rate K_n+1:

K_n+1=α K_n+(1-α)K_n-1+(1-α)²K_n-2 (9)

Heat is triggered in temperature field:

Q^pre=△ T_maxK_n+1(10)；

Step 6): start electromagnetic field, determine electromagnetic field time step according to load discretization errort∈(t_n-1,t_n) time, when When loading matrix P uses linear interpolation, equivalent load uses slope to load, the error of the discrete generation of load be such as formula (13),

$e_s^n=\frac{1}{\mathrm{\&Delta;}t}\left|{\&Integral;}_{t_{n-1}}^{t_n}\right|f_s\left(t\right)|dt-\frac{\left|f_s\left(t_{n-1}\right)\right|+\left|f_s\left(t_n\right)\right|}{2}\mathrm{\&Delta;}t|---\left(13\right)$

Formula (13) is used trapezoid formula integration, obtains discretization error and approximate such as formula (14)；

$e_s^n=\frac{1}{\mathrm{\&Delta;}t}\left|\frac{f_s^{\&prime;\&prime;}\left(\mathrm{\&xi;}\right)}{12}{\left(\mathrm{\&Delta;}t\right)}^3\right|=\left|\frac{f_s^{\&prime;\&prime;}\left(\mathrm{\&xi;}\right)}{12}{\left(\mathrm{\&Delta;}t\right)}^2\right|\&Proportional;{\left(\mathrm{\&Delta;}t\right)}^2---\left(14\right)$

According to formula (14), load discretization error is approximately proportional to (△ t)², next step size computation can be able to be divided into following two steps:

(a), step-ahead prediction: calculate error according to the n-th step, it was predicted that the (n+1)th step-length △ t¹ _n+1,

${\mathrm{\&Delta;t}}_{n+1}^1=K_{SF}^1{\mathrm{\&Delta;t}}_n{\left(\frac{e_{tolerance}}{e_s^n}\right)}^{\frac{1}{2}}---\left(15\right)$

In formula:For safety coefficient,e_toleranceFor allowing maximum error；It is the discrete of the interior generation of the n-th time step Error,

B (), step-length correct: judge as the (n+1)th time step △ t¹ _n+1Whether produced error meets e^k+1<e_tolerance, as discontented Foot uses (8) to be modified iterative computation, until meeting e^k<e_tolerance,

${\mathrm{\&Delta;t}}_{n+1}^1=K_{SF}^2{\mathrm{\&Delta;t}}_{n+1}^1{\left(\frac{e_{tolerance}}{e^{k+1}}\right)}^{\frac{1}{2}}---\left(16\right)$

In formula:For safety coefficient,

Step 7): determine electromagnetic field time step according to response characteristic valueHughes proposes according to response characteristic λ_rValue determines Calculate step-length △ t stabilization time_n+1Method^[16], define △ t_n+1λ is concussion restrictive condition, as △ t_n+1λ > > 1 time system be in Concussion state, for ensureing that the desirable maximum step-length of computational stability need to meet:

${\mathrm{\&Delta;t}}_{n+1}^2{\mathrm{\&lambda;}}_r=f---\left(17\right)$

${\mathrm{\&lambda;}}_r=\frac{{\mathrm{\&Delta;u}}_n^T{\mathrm{K\&Delta;u}}_n}{{\mathrm{\&Delta;u}}_n^{T}C\mathrm{\&Delta;}u}---\left(18\right)$

In formula: f < 1, f is the coefficient of stability；λ_rFor response characteristic value；△u_nFor t_n-1To t_nThe change of time period field amount u；

Step 8): electromagnetic field is at t_nMoment discrete time step is:

${\mathrm{\&Delta;t}}_{n+1}^E=min({\mathrm{\&Delta;t}}_{n+1}^1,{\mathrm{\&Delta;t}}_{n+1}^2)---\left(19\right)$

Step 9): coupling time judges: work as T_n+1Moment electromagnetic field accumulation heat meets following condition for the moment, T_n+1For electromagnetism- Temperature field coupling time point, start-up temperature field calculates；

(a) unit maximum heat change reach step 5) in unit prediction heat threshold time automatic start-up temperature field calculate:

$max\left(Q^i\right)\&GreaterEqual;Q_{n+1}^{i\_pre}---\left(20\right)$

(b) overall heat change reach step 5) in unit prediction heat threshold time automatic start-up temperature field calculate:

$\underset{i=1}{\overset{n}{\&Sigma;}}{Q}^i\&GreaterEqual;{\mathrm{\&beta;Q}}^{pre}---\left(21\right)$

Step 10): start-up temperature field calculates, and calculates temperature field adaptive time-stepTemperature field adaptive time-step meter Calculate same step 6), step 7), step 8)；

Step 11): when the Temperature calculating time arrive step 9) in coupling time T_n+1Time, stop Temperature calculating, and update electricity Magnetic field node temperature；

Step 12): iterate step 4)～step 10), until calculating total time T_total。