Multi-physics field simulation method based on mixing of time domain discontinuous Galerkin and finite element

Abstract

The invention discloses a multi-physical field simulation method based on time domain discontinuous Galerkin and finite element mixing, and mainly solves the problems of low calculation speed and high memory consumption of electromagnetic-thermal-mechanical multi-physical simulation. Comprising the following steps: 1) setting the total time step number of simulation and initializing; 2) performing electromagnetic simulation by using a time domain discontinuous Galerkin method to obtain electromagnetic field distribution; 3) performing thermal simulation by using a finite element method based on an electromagnetic simulation result to obtain temperature distribution; 4) correcting the relationship between stress and strain according to the temperature variation of thermal simulation, and performing force simulation by using a finite element method to obtain displacement to obtain a deformation grid; and 5) electromagnetic simulation of the next time step is carried out based on the deformation grid until the total time step number is completely completed, and the multi-physics field simulation task is ended. According to the invention, the speed of multi-physics field simulation can be effectively improved, and the memory consumption is reduced, so that the capability of multi-physics field simulation is improved.

Description

Technical Field

[0001] The present invention belongs to the field of electrical technology and further relates to computational electromagnetics. Specifically, it is a multi-physics field simulation method based on a mixture of time-domain discontinuous Galerkin and finite element, which can be used for accurate and efficient analysis of electromagnetic, thermal and force multi-physics field coupling problems in the fields of wireless communications, integrated circuits, microwave circuits, antenna design, precision guidance, etc. Background Art

[0002] With the rapid development of wireless communications, integrated circuits, microwave circuits, antenna design, precision guidance and other fields, the coupling problem of multiple physical fields such as electromagnetic fields, temperature fields, and stress fields, centered on electromagnetic fields, has become increasingly prominent. The actual measurement of the above electromagnetic-thermal-force multi-physical field coupling problems has the disadvantages of high cost, long cycle, and the results are easily affected by the test environment. Therefore, electromagnetic-thermal-force multi-physics simulation technology has become a necessary means to study the effects of multiple physical fields. Multi-physics simulation is a typical multidisciplinary cross-disciplinary technology that requires relevant knowledge in physics, mathematics, computer science, and engineering applications. It mainly uses numerical algorithms to solve the control equations of different physical fields, and realizes multi-physical field coupling simulation by establishing coupling relationships between different physical fields.

[0003] At present, the mainstream electromagnetic-thermal-force multi-physics simulation method is the finite element method. For example, COMSOL Multiphysics, a multi-physics numerical simulation software widely used worldwide, is developed based on the finite element method. It uses the finite element method to simulate electromagnetic, thermal, and force single physical fields, and then establishes the coupling relationship between electromagnetic simulation, thermal simulation, and force simulation, and finally realizes electromagnetic-thermal-force multi-physics simulation. The advantages of the finite element method mainly include the ability to use tetrahedral meshes to divide geometric structures, the ability to approximate arbitrary geometric structures, the ability to use high-order basis functions to improve calculation accuracy, and the ability to analyze complex media problems; its disadvantage is that it solves large global matrix equations. The scale of the equation is related to the amount of grids. As the scale of the problem to be solved increases, the scale of the matrix equation also increases sharply, resulting in great difficulties in solving the matrix equations whether using direct solutions or iterative solutions. For example, in the patent document with publication number CN112001101A and titled "GIL electro-magnetic-thermal-fluid-force multi-physics field coupling simulation modeling method based on finite element", a method is specifically disclosed that can calculate the distribution of electric field, magnetic field, thermal field, flow field, and thermal expansion displacement field during the operation of GIL, and simulate the electro-magnetic-thermal-fluid-force effects during the actual operation of GIL. Although this method can improve the calculation accuracy to a certain extent, it still has problems of slow calculation speed and high memory consumption. Summary of the invention

[0004] The purpose of the present invention is to address the deficiencies of the above-mentioned prior art, mainly to address the difficulties of electromagnetic-thermal-force multi-physics simulation, and to propose a multi-physics field simulation method based on a mixture of time-domain discontinuous Galerkin and finite element. The time-domain discontinuous Galerkin algorithm is used for electromagnetic simulation, and the finite element algorithm is used for thermal and force simulation. Then, a coupling relationship among electromagnetic simulation, thermal simulation, and force simulation is established, and finally electromagnetic-thermal-force multi-physics field simulation is realized. This method can effectively improve the speed of multi-physics field simulation and reduce memory consumption, thereby improving the capability of multi-physics field simulation.

[0005] The present invention achieves the above-mentioned purpose by the following specific steps:

[0006] (1) Set the total time step number of transient electromagnetic-thermal-mechanical multi-physics field simulation to N, and the time step of the current simulation is recorded as the i-th time step, i = 1, 2, ..., N; let i = 1 to start stepping;

[0007] (2) For the i-th time step, the time-domain discontinuous Galerkin method is used to perform electromagnetic simulation to obtain the electromagnetic field distribution of the entire solution domain;

[0008] (3) Based on the results of electromagnetic simulation, the finite element method is used to perform thermal simulation at the current time step to obtain the temperature distribution of the entire solution domain;

[0009] (4) Determine the temperature change ΔT of the object based on the temperature distribution obtained by thermal simulation. After correcting the relationship between stress and strain, use the finite element method to perform force simulation at the current time step to obtain the displacement of the entire solution domain, and superimpose the displacement on the original node coordinates to obtain the deformation mesh;

[0010] (5) Determine whether i is less than N. If so, set i=i+1 and return to step (2) to perform electromagnetic simulation of the next time step based on the deformed mesh obtained by force simulation. Otherwise, continue to step (6);

[0011] (6) End the simulation process and complete the transient electromagnetic-thermal-mechanical multi-physics field simulation task.

[0012] Compared with the prior art, the present invention has the following advantages:

[0013] First, the present invention uses the time-domain discontinuous Galerkin method instead of the finite element method for electromagnetic simulation, so there is no need to solve the global matrix equation, only each unit matrix equation needs to be solved, which is naturally suitable for parallel processing, thereby effectively improving the electromagnetic simulation speed;

[0014] Second, since the invention uses the time-domain discontinuous Galerkin method for electromagnetic simulation, considering that the amount of calculation for thermal and force simulation is small, the finite element algorithm is still chosen to perform thermal simulation and force simulation, thereby improving the calculation accuracy of thermal and force simulation;

[0015] Third, the present invention, under the premise of comprehensively considering the respective characteristics of electromagnetic, thermal and force, adopts different methods to simulate them respectively, and then establishes the coupling relationship between multiple different physical fields of electromagnetic simulation, thermal simulation and force simulation, to realize electromagnetic-thermal-force multi-physical field simulation based on the time-domain discontinuous Galerkin and finite element hybrid algorithm architecture, so as to achieve the purpose of improving the multi-physical field simulation speed and reducing memory consumption. BRIEF DESCRIPTION OF THE DRAWINGS

[0016] Figure 1 It is a flow chart for realizing the present invention;

[0017] Figure 2 A schematic diagram of the electromagnetic-thermal-force multi-physics field coupling relationship provided in an embodiment of the present invention;

[0018] Figure 3 A schematic diagram of a microstrip matching network for a MESFET microwave power amplifier provided in an embodiment of the present invention;

[0019] Figure 4 A top view of the heat source position of a MESFET microwave power amplifier provided in an embodiment of the present invention;

[0020] Figure 5 The figure is a comparison diagram of the simulation results of the method of the present invention and the COMSOL software; wherein (a) is the displacement distribution, and (b) is the port voltage. DETAILED DESCRIPTION

[0021] The present invention is further described in detail below in conjunction with the accompanying drawings.

[0022] Example 1: Reference Figure 1 The implementation steps of the present invention are as follows:

[0023] Step 1. Set the total time step number of transient electromagnetic-thermal-mechanical multi-physics simulation to N, and the time step of the current simulation is recorded as the i-th time step, i = 1, 2, ..., N; let i = 1 to start stepping;

[0024] Step 2. For the i-th time step, use the time domain discontinuous Galerkin method to solve the following Maxwell curl equation for electromagnetic simulation to obtain the electromagnetic field distribution of the entire solution domain;

[0025]

[0026]

[0027] Where, E is the electric field strength, H is the magnetic field strength, ε is the dielectric constant, μ is the magnetic permeability, σ is the electrical conductivity, and t is the time variable. is the Hamiltonian operator.

[0028] Step 3. Based on the results of electromagnetic simulation, use the finite element method to perform thermal simulation at the current time step to obtain the temperature distribution of the entire solution domain. The specific implementation steps are as follows:

[0029] (3.1) Calculation of conductor loss P based on electromagnetic simulation results c and dielectric loss P d :

[0030] P c =σ|E 2 ,

[0031]

[0032] Where D represents the electric displacement vector;

[0033] (3.2) The heat source term Q is obtained according to the following formula:

[0034] Q=P c +P d ,;

[0035] (3.3) The finite element method is used to perform thermal simulation of the current time step. Specifically, the finite element method is used to solve the following heat conduction equation to obtain the temperature distribution:

[0036]

[0037] Where ρ is the material density, C ρ is the specific heat capacity, κ is the thermal conductivity, is the Hamiltonian operator, T' is the transient temperature distribution, Q represents the heat source term, and t is the time variable.

[0038] Step 4. Determine the temperature change ΔT of the object based on the temperature distribution obtained by thermal simulation. After correcting the relationship between stress and strain, use the finite element method to perform force simulation at the current time step to obtain the displacement of the entire solution domain, and superimpose the displacement on the original node coordinates to obtain the deformed mesh.

[0039] The correction of the relationship between stress and strain is achieved according to the following formula:

[0040] σ=Cε+σ′,

[0041] Among them, σ is the stress vector composed of normal stress and shear stress, C is the elastic coefficient matrix, ε is the strain vector composed of normal strain and shear strain, σ′=[-βΔT,-βΔT,-βΔT,0,0,0] T , thermal stress coefficient E' is the elastic modulus, α is the thermal expansion coefficient, [] T Represents a transpose operation.

[0042] The force simulation of the current time step is performed by using the finite element method, specifically, the displacement is obtained by solving the following elastic mechanics equation using the finite element method:

[0043]

[0044] In the above formula, ρ is the material density, μ′ is the damping coefficient, f is the body force vector, L is the differential operator matrix, and u = [u, v, w] T It is a displacement vector composed of three components u, v, and w in the rectangular coordinate system.

[0045] The differential operator matrix L and the elastic coefficient matrix C are expressed as follows:

[0046]

[0047] Among them, x, y, and z represent the three directional components in the three-dimensional coordinate system.

[0048]

[0049] Among them, the Lamé constant Shear modulus E' is the elastic modulus and υ is the Poisson's ratio.

[0050] Step 5. Determine whether i is less than N. If so, set i=i+1 and return to step 2 to perform electromagnetic simulation for the next time step based on the deformed mesh obtained by force simulation. Otherwise, continue to step 6.

[0051] Step 6. End the simulation process and complete the transient electromagnetic-thermal-mechanical multi-physics field simulation task.

[0052] Example 2: Reference Figure 2 The overall implementation steps of the present invention are the same as those of the first embodiment. Now, a specific example is given to further describe the implementation process of the present invention in detail:

[0053] 1) Use the time domain discontinuous Galerkin method to solve the following Maxwell curl equation:

[0054]

[0055] In the above formula, E is the electric field strength, H is the magnetic field strength, ε is the dielectric constant, μ is the magnetic permeability, σ is the electrical conductivity, and t is the time variable. <1> Formula and <2> The electromagnetic field distribution of the entire solution domain can be obtained after the formula, thus realizing electromagnetic simulation;

[0056] 2) Use the finite element method to solve the following heat conduction equation:

[0057]

[0058] In the above formula, ρ is the material density, Cρ is the specific heat capacity, κ is the thermal conductivity, T is the transient temperature distribution, and Q is the heat source term. <3> The temperature distribution of the entire solution domain can be obtained by the formula, thus realizing thermal simulation;

[0059] 3) Use the finite element method to solve the following elastic mechanics equations:

[0060]

[0061] ε=L T u <5>

[0062] σ=Cε <6>

[0063] In the above formula, ρ is the material density, μ′ is the damping coefficient, and f = [f x ,f y ,f z ] T is the body force vector, u=[u,v,w] T is the displacement vector. σ=[σ x ,σ y ,σ z ,τ xy ,τ yz ,τ zx ] T is the stress vector, which is composed of the normal stress σ x , σ y , σ z and shear stress τ xy , τ yz , τ zx ε=[ε x ,ε y ,ε z ,γ xy ,γ yz ,γ zx ] T is the strain vector, which is composed of the positive strain ε x , ε y , ε z and shear strain γ xy , γ yz , γ xz L is the differential operator matrix, and

[0064]

[0065] C is the elastic coefficient matrix. When the material is isotropic, C can be expressed as:

[0066]

[0067] Among them, the Lamé constant Shear modulus E is the elastic modulus and υ is the Poisson's ratio.

[0068] Solution <4> Mode, <5> Mode, <6> The displacement of the entire solution domain can be obtained by solving the equation group composed of the formula. The displacement is superimposed on the original node coordinates to obtain the deformed mesh.

[0069] 4) If Figure 1 As shown, the coupling relationship among electromagnetic simulation, thermal simulation and force simulation is established.

[0070] First, the coupling from electromagnetic simulation to thermal simulation is introduced. After the electromagnetic simulation, the conductor loss P is calculated using the following two formulas: c and dielectric loss P d :

[0071] P c =σ|E| 2 <9>

[0072]

[0073] Where D is the electric displacement vector, then P c +P d It is used as the heat source term in the heat conduction equation, and the temperature distribution is obtained by thermal simulation.

[0074] The coupling from thermal simulation to electromagnetic simulation is achieved through the influence of temperature changes on dielectric constant and conductivity. During electromagnetic analysis, the dielectric constant and conductivity can be updated according to the current temperature.

[0075] The influence of thermal simulation on force simulation works through the thermal expansion coefficient of the material. Thermal simulation obtains the temperature change ΔT of the object. The temperature change will cause the expansion effect. Considering the strain caused by the temperature change, the relationship between stress and strain shown in formula (6) needs to be corrected:

[0076] σ=Cε+σ′ <11>

[0077] Among them, σ′=[-βΔT,-βΔT,-βΔT,0,0,0] T , thermal stress coefficient α is the thermal expansion coefficient and ΔT is the temperature change.

[0078] use <4> Mode, <5> Mode, <11> Thermal expansion effect analysis can be achieved by performing force simulation.

[0079] The coupling of force simulation to electromagnetic simulation and thermal simulation is achieved through deformable mesh, that is, the deformable mesh obtained by mechanical simulation is used to perform electromagnetic simulation and thermal simulation.

[0080] In the electromagnetic, thermal and force multi-physics field simulation, the unknown quantity of electromagnetic simulation is related to the number of grid edges, and the calculation amount of thermal and force simulation is related to the number of grid nodes. Since the number of edges is much larger than the number of nodes, the calculation amount of electromagnetic simulation is much larger than the calculation amount of thermal and force simulation. Therefore, the present invention uses the time-domain discontinuous Galerkin method instead of the finite element method to perform electromagnetic simulation. The time-domain discontinuous Galerkin method does not need to solve the global matrix equation, but only needs to solve each unit matrix equation. It is naturally suitable for parallelization and can effectively improve the electromagnetic simulation speed. Since the calculation amount of thermal and force simulation is relatively small, the finite element algorithm is still used for thermal and force simulation, and then the coupling relationship of electromagnetic simulation, thermal simulation and force simulation is established, and finally an electromagnetic-thermal-force multi-physics field simulation method based on the time-domain discontinuous Galerkin and finite element hybrid algorithm architecture is realized. This method can effectively improve the speed of multi-physics field simulation and reduce memory consumption, thereby improving the ability of multi-physics field simulation.

[0081] The effect of the present invention is further described below in conjunction with simulation experiments.

[0082] This experiment uses the multi-physics field simulation method proposed in the present invention and the traditional electromagnetic-thermal-mechanical multi-physics field simulation method based on finite elements to simulate the same MESFET microwave power amplifier example, compares the calculation time and memory consumption of the two, and verifies the effectiveness of the present invention.

[0083] Microstrip matching network for MESFET microwave power amplifier Figure 3 As shown in Figure 1, a total of four ports are defined in the matching network: Port a, Port b, Port G, and Port D, and the MESFET is connected between Port G and Port D. Figure 4 As shown, in order to simulate the actual operating temperature of the MESFET microwave power amplifier, four rectangular blocks of 2.4mm×1.2mm×1.2mm are placed on the dielectric bottom plate of the MESFET microwave power device along the x-axis direction as heat sources for thermal simulation, and the total power is set to 0.05W. The electromagnetic-thermal-force multi-physics field simulation method based on the time-domain discontinuous Galerkin and finite element hybrid algorithm architecture proposed in the present invention and COMSOL software are used to analyze the influence of thermal deformation of the MESFET microwave power amplifier on its electromagnetic performance. The COMSOL software adopts the electromagnetic-thermal-force multi-physics field simulation method based on traditional finite elements. The simulation results are shown in FIG. Figure 5 As shown, the two are well matched, proving the correctness of the present invention.

[0084] In addition, in this example, the memory consumption and calculation time of the present invention are 2044MB and 9484 seconds respectively, and the memory consumption and calculation time of COMSOL software are 3600MB and 12063s respectively. The memory consumption and calculation time of the present invention are significantly less than those of COMSOL software, which proves the correctness and effectiveness of the present invention.

[0085] Parts of the present invention that are not described in detail belong to common knowledge among those skilled in the art.

[0086] The above description is only a preferred embodiment of the present invention and is not intended to limit the present invention. Obviously, for professionals in this field, after understanding the content and principles of the present invention, they may make various modifications and changes in form and details without departing from the principles and structures of the present invention. However, these modifications and changes based on the ideas of the present invention are still within the scope of protection of the claims of the present invention.

Claims

1. A multi-physics field simulation method based on time domain discontinuous Galerkin and finite element hybrid, characterized in that: The steps include: (1) Set the total time step number of transient electromagnetic-thermal-mechanical multi-physics field simulation to N, and the time step of the current simulation is recorded as the i-th time step, i = 1, 2, ..., N; let i = 1 to start stepping; (2) For the i-th time step, the time-domain discontinuous Galerkin method is used to perform electromagnetic simulation to obtain the electromagnetic field distribution of the entire solution domain; (3) Based on the results of electromagnetic simulation, the finite element method is used to perform thermal simulation at the current time step to obtain the temperature distribution of the entire solution domain; (4) Determine the temperature change ΔT of the object based on the temperature distribution obtained by thermal simulation. After correcting the relationship between stress and strain, use the finite element method to perform force simulation at the current time step to obtain the displacement of the entire solution domain, and superimpose the displacement on the original node coordinates to obtain the deformation mesh; (5) Determine whether i is less than N. If so, set i=i+1 and return to step (2) to perform electromagnetic simulation of the next time step based on the deformed mesh obtained by force simulation. Otherwise, continue to step (6); (6) End the simulation process and complete the transient electromagnetic-thermal-mechanical multi-physics field simulation task.

2. The method according to claim 1, characterized in that: The electromagnetic simulation is performed using the time-domain discontinuous Galerkin method in step (2), specifically, the time-domain discontinuous Galerkin method is used to solve the following Maxwell curl equation to obtain the electromagnetic field distribution: Among them, E is the electric field intensity, H is the magnetic field intensity, ε is the dielectric constant, μ is the magnetic permeability, σ is the electrical conductivity, t is the time variable, and ▽ is the Hamiltonian operator.

3. The method according to claim 1, characterized in that: The thermal simulation of the current time step is performed using the finite element method in step (3), specifically, the finite element method is used to solve the following heat conduction equation to obtain the temperature distribution: Where ρ is the material density, C ρ is the specific heat capacity, κ is the thermal conductivity, is the Hamiltonian operator, T' is the transient temperature distribution, Q represents the heat source term, and t is the time variable.

4. The method according to claim 3, characterized in that: The heat source term is obtained according to the following formula: Q=P c +P d , Among them, P c is the conductor loss, P d is the dielectric loss.

5. The method according to claim 4, characterized in that: The conductor loss and dielectric loss are based on the results of electromagnetic simulation and are calculated as follows: P c =σE 2 , Where D represents the electric displacement vector.

6. The method according to claim 1, characterized in that: Step (4) corrects the relationship between stress and strain according to the following formula: σ=Cε+σ′, Among them, σ is the stress vector composed of normal stress and shear stress, C is the elastic coefficient matrix, ε is the strain vector composed of normal strain and shear strain, σ′=[-βΔT,-βΔT,-βΔT,0,0,0] T , thermal stress coefficient E' is the elastic modulus, α is the thermal expansion coefficient, [] T Represents a transpose operation.

7. The method according to claim 6, characterized in that: The force simulation of the current time step is performed using the finite element method in step (4), specifically, the displacement is obtained by solving the following elastic mechanics equation using the finite element method: e=L T you, In the above formula, ρ is the material density, μ′ is the damping coefficient, f is the body force vector, L is the differential operator matrix, and u = [u, v, w] T It is a displacement vector composed of three components u, v, and w in the rectangular coordinate system.

8. The method according to claim 6, characterized in that: The elastic coefficient matrix C is expressed as follows: Among them, the Lamé constant Shear modulus E' is the elastic modulus and υ is the Poisson's ratio.

9. The method according to claim 7, characterized in that: The differential operator matrix L is expressed as follows: Among them, x, y, and z represent the three directional components in the three-dimensional coordinate system.

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