PCE-FDTD-based quantitative analysis method for uncertainty of electromagnetic-thermal multi-physics field of chip

Abstract

The invention discloses a PCE-FDTD-based quantitative analysis method for uncertainty of an electromagnetic-thermal multi-physical field of a chip. According to the method, the calculation cost of uncertainty quantitative analysis caused by randomness of input parameters such as material attributes and geometric dimensions in the electromagnetic-thermal coupling system is greatly reduced, and the engineering bottlenecks of too long time consumption and huge resource consumption caused by the fact that a traditional Monte Carlo method needs to execute massive deterministic simulation are overcome. The high-precision statistical characteristics, including the mean value and the variance, of the output response (such as a temperature field) can be synchronously obtained through single-time or finite-time simulation, and the analysis efficiency is improved by dozens of times to hundreds of times on the premise that the result reliability is guaranteed. An efficient and feasible numerical analysis tool is provided for reliability evaluation, performance prediction and robust design of electrothermal coupling systems such as radio frequency integrated circuits and high-power devices.

Description

Technical Field

[0001] This technology belongs to the field of computational electromagnetics and multiphysics simulation technology, specifically involving a chip electromagnetic-thermal multiphysics uncertainty quantification analysis method based on PCE-FDTD. Background Technology

[0002] With the continuous evolution and innovation of radio frequency integrated circuit technology, highly integrated and miniaturized radio frequency integrated circuits exhibit significant multi-physics field effects. Multi-physics co-simulation is an important tool in circuit design and has attracted widespread attention from scholars in the field of computational electromagnetics. FDTD (Finite-Difference Time-Domain) is a classic numerical method for solving Maxwell's equations and is widely used in electromagnetic field simulation. In scenarios involving thermal effects (such as microwave heating, laser processing, and heat dissipation of high-power devices), the electromagnetic parameters of materials (such as dielectric constant and conductivity) are often functions of temperature, forming electromagnetic-thermal coupling.

[0003] Existing coupled FDTD methods (such as those incorporating heat conduction equations for simultaneous solving) typically rely on deterministic parameters for simulation. This assumes that all input parameters, including material properties, geometric dimensions, and excitation sources, are precisely known. However, in reality, input parameters inherently possess uncertainties (such as measurement errors in material properties, manufacturing tolerances, and environmental fluctuations). The Monte Carlo (MC) method, a stochastic simulation method based on probability and statistics, has become the standard means of quantifying uncertainties in numerical models across multiple domains. In computational electromagnetics, researchers integrate it with FDTD to solve electromagnetic problems involving stochastic media, rough surfaces, or geometric variations. This MC-FDTD hybrid strategy fundamentally overcomes the inherent limitations of traditional FDTD methods in handling stochastic problems. However, its application faces a major bottleneck: to obtain reliable statistical results, hundreds or even thousands of independent FDTD simulations must be performed. The resulting enormous computational load often makes this method impractical for solving real-world problems due to its excessive time consumption. The Stochastic Finite-Difference Time-Domain (S-FDTD) method directly handles random variables within the FDTD equations, introducing randomness into the FDTD calculation. This method can obtain the statistical properties of the electromagnetic field in a single simulation. However, S-FDTD introduces errors when estimating correlation coefficients, affecting the accuracy of the results, and may have limitations when dealing with complex random media.

[0004] Therefore, this invention proposes an uncertainty quantification technique based on PCE-FDTD. This method establishes a coupling mechanism between the heat conduction equation and PCE, constructing a high-precision electromagnetic-thermal coupling model that can directly embed random parameters. Compared to traditional MC-FDTD, which relies on repeated sampling and extensive computation, this method requires only one simulation to simultaneously obtain the statistical characteristics of the output response (such as mean and variance), and avoids errors in correlation coefficient estimation. It significantly improves computational efficiency and reduces resource consumption while maintaining accuracy. This technology can provide crucial support for the reliability analysis and design of electrothermal coupling systems such as RF integrated circuits and high-power devices. Summary of the Invention

[0005] To address the limitations of existing uncertainty quantification techniques in electromagnetic-thermal coupling systems, such as insufficient accuracy, high computational cost, and incomplete statistical characterization, especially the traditional Monte Carlo method which requires thousands of deterministic simulations for uncertainty quantification, resulting in extremely high computational costs, this invention aims to overcome these shortcomings by providing an efficient and high-precision uncertainty quantification method. This method can quantitatively analyze the impact of the randomness of input parameters on the statistical properties (such as mean and variance) of output results in electromagnetic-thermal coupling systems within an acceptable computational cost.

[0006] To achieve the above objectives, this invention provides a method for quantifying uncertainties in the electromagnetic-thermal multiphysics field of a chip based on PCE-FDTD, comprising the following steps: S1. Based on the uncertain input parameters and their probability distribution in the electromagnetic-thermal coupling system, construct an electromagnetic-thermal coupling simulation model containing random parameters, and determine the basis functions and total number of terms of the polynomial chaotic expansion PCE. S2. Based on the electromagnetic-thermal coupling simulation model, the heat conduction equation is discretized using the finite-difference time-domain (FDTD) method, and the random thermal field quantity is expanded using the basis functions of the PCE to obtain the PCE representation of the thermal field quantity. S3. Based on the PCE representation of the thermal field and the PCE representation of the random input parameters, the updated equation of the PCE coefficient is obtained through Galerkin projection, and then solved by combining the electromagnetic power dissipation density and the convective boundary conditions to obtain the updated PCE coefficient. S4. Calculate the mean and variance of the output field based on the updated PCE coefficients.

[0007] Preferably, in step S1, the step of constructing an electromagnetic-thermal coupling simulation model containing random parameters includes: defining the thermal conductivity and specific heat capacity of the medium as random input parameters, and specifying a normal distribution or a uniform distribution for each random parameter.

[0008] Preferably, in step S1, the step of determining the basis functions of the polynomial chaotic expansion PCE includes: selecting the corresponding orthogonal polynomial basis according to the distribution type of the standard random variable, wherein Hermite polynomial is selected when the input is a Gaussian distribution, and Legendre polynomial is selected when the input is a uniform distribution.

[0009] Preferably, in step S2, the step of obtaining the PCE representation of the thermal field includes: in, i , j , k express x, y, z direction; n Indicates a time step; Indicates the time step; , , Indicates the spatial step size.

[0010] Using orthogonal polynomial basis functions Unfold the stochastic thermal field for each Yee unit and time step: in, Indicates the first l Coefficients of the temperature expansion term; P Indicates the number of terms expanded; Represents a variable.

[0011] Preferably, in step S3, obtaining the update equation for the PCE coefficients through Galerkin projection includes: projecting both sides of the discretized heat conduction equation onto the polynomial basis functions, and using the orthogonality condition of the basis functions to obtain the update equation for the PCE coefficients.

[0012] Preferably, in step S3, the method for solving the problem by combining the electromagnetic power dissipation density includes: calculating the electromagnetic power dissipation density Q based on the electric field distribution E, wherein the relationship between Q and E is as follows: in, This indicates electrical conductivity.

[0013] Preferably, in step S4, the step of calculating the mean and variance of the output field based on the updated PCE coefficients includes: in, Represents the coefficient of the lowest-order temperature expansion term; Represents the square of a polynomial expansion; μ This represents the mean; Indicates variance; P Indicates the number of terms expanded; A temperature function representing the effect of variables; Indicates the first l Coefficients of the temperature expansion term.

[0014] Compared with the prior art, the beneficial effects of the present invention are as follows: This method significantly reduces the computational cost of uncertainty quantification analysis in electromagnetic-thermal coupling systems due to the randomness of input parameters such as material properties and geometric dimensions. It overcomes the engineering bottleneck of traditional Monte Carlo methods, which require massive deterministic simulations, resulting in excessive time consumption and resource demands. High-precision statistical characteristics of the output response (such as the temperature field), including mean and variance, can be obtained simultaneously through a single or finite number of simulations, improving analysis efficiency by tens to hundreds of times while ensuring the reliability of the results. This provides an efficient and feasible numerical analysis tool for reliability assessment, performance prediction, and robust design of electrothermal coupling systems such as RF integrated circuits and high-power devices. Attached Figure Description

[0015] To more clearly illustrate the technical solution of the present invention, the drawings used in the embodiments are briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0016] Figure 1 This is a schematic diagram of a silicon-based rectangular block model according to an embodiment of the present invention; Figure 2 This is a three-dimensional temperature distribution diagram of the silicon-based rectangular block model in an embodiment of the present invention at a simulation time t=160s; wherein, Figure 2 In the table, (a) represents the analytical solution; (b) represents the solution when PCE, D=1; and (C) represents the solution when PCE, D=2.

[0017] Figure 3 This is a schematic diagram comparing the mean values ​​of the silicon-based rectangular block model MC-FDTD and PCE-FDTD (D=1, D=2) in an embodiment of the present invention; where D represents the highest order of PCE. Figure 4 This is a schematic diagram comparing the variances of the silicon-based rectangular block models MC-FDTD and PCE-FDTD (D=1, D=2) in an embodiment of the present invention. Figure 5 This is a simulation model diagram of the microchip according to an embodiment of the present invention; Figure 5 In the diagram, (a) represents the overall view; (b) represents... yz Cross-sectional view; (c) indicates xz Cross-sectional view; Figure 6 This is a schematic diagram comparing the mean values ​​of the microchip models MC-FDTD and PCE-FDTD (D=1, D=2) in an embodiment of the present invention. Figure 7 This is a schematic diagram comparing the variances of the microchip models MC-FDTD and PCE-FDTD (D=1, D=2) in an embodiment of the present invention. Detailed Implementation

[0018] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0019] To make the above-mentioned objects, features and advantages of the present invention more apparent and understandable, the present invention will be further described in detail below with reference to the accompanying drawings and specific embodiments.

[0020] Example 1 This embodiment provides a chip electromagnetic-thermal multiphysics uncertainty quantification analysis method based on PCE-FDTD, the steps of which include: S1. Based on the uncertain input parameters and their probability distribution in the electromagnetic-thermal coupling system, construct an electromagnetic-thermal coupling simulation model containing random parameters, and determine the basis functions and total number of terms of the polynomial chaotic expansion PCE.

[0021] Define random input parameters that affect the system, such as the thermal conductivity k and specific heat capacity Cp of the medium, and assign a suitable probability distribution to each random parameter. Derive the iterative formula for the Finite-Difference Time-Domain (FDTD) method for the thermal field using Polynomial Chaos Expansion (PCE). First, verify the effectiveness of the PCE-FDTD method for the thermal field using a cubic model, and then verify the computational effectiveness of the PCE-FDTD method combined with electromagnetic-thermal coupling after inputting random medium parameters using a microchip model.

[0022] Furthermore, the physical structure of the simulation region is defined, including the precise dimensions, shape, and spatial location of each component. Materials are assigned to each region; for fixed materials, definite electromagnetic and thermal parameters are provided. For materials with uncertain input, the functional relationships or discrete data tables of their electromagnetic or thermal parameter variations must be explicitly defined.

[0023] Furthermore, all possible input parameters with uncertainty are systematically identified and listed, and appropriate probability distribution functions and parameters are assigned to each identified random parameter.

[0024] Furthermore, based on the distribution type of the standard random variable ξ, the corresponding orthogonal polynomial basis is automatically selected (in this embodiment, Hermite polynomial corresponds to Gaussian distribution, and Legendre polynomial corresponds to uniform distribution), and the total number of terms P of the PCE is determined. The standard random variable ξ is a user-defined variable that influences the input parameters; its range is from 1 to -1, corresponding to Gaussian or uniform distributions, etc.

[0025] Furthermore, define the source type, thermal boundary conditions, and electromagnetic boundary conditions, and correctly incorporate them into the algorithm for calculation.

[0026] Furthermore, initial values ​​for all PCE coefficient fields are set, and PCE coefficients are calculated and stored appropriately using MATLAB based on the PCE representations of random input parameters and temperature field PCE representations.

[0027] S2. Based on the electromagnetic-thermal coupling simulation model, the heat conduction equation is discretized using the finite-difference time-domain (FDTD) method, and the random thermal field quantity is expanded using the basis functions of the PCE to obtain the PCE representation of the thermal field quantity.

[0028] This step mainly demonstrates the principle of the electromagnetic-thermal multiphysics uncertainty quantification model based on PCE-FDTD. In thermal analysis, the basic equation for simulating temperature transfer is the heat conduction equation: (1) in, ρ It is the material density. Cp Specific heat capacity k It is thermal conductivity. T Indicates temperature. Q Indicates a heat source.

[0029] By applying the FDTD method to transform it, formula (1) becomes: (2) in, i , j , k express x, y, z direction; n Indicates a time step; Indicates the time step; , , Indicates the spatial step size.

[0030] Using orthogonal polynomial basis functions Unfold the stochastic thermal field for each Yee unit and time step: (3) in, Indicates the first l Coefficients of the temperature expansion term; P Indicates the number of terms expanded; Represents a variable.

[0031] Among them P It is obtained from the following: (4) in, N Where is the number of variables, and D is the order of the highest polynomial in the expansion. The basis functions of a multivariate polynomial chaos can be expressed as: (5) in, yes The unary orthogonal basis in Is it related to the order of expansion? The corresponding expansion parameters. Although any suitable orthogonal function can be used, it can be proven that the optimal polynomial basis... Distribution of random variables that depends on the assumption This association is known as the Wiener-Askey scheme, where Gaussian-distributed inputs are associated with Hermite polynomials, while uniformly distributed inputs are associated with Legendre polynomials.

[0032] S3. Based on the PCE representation of the thermal field and the PCE representation of the random input parameters, the updated equation of the PCE coefficient is obtained through Galerkin projection, and then solved by combining the electromagnetic power dissipation density and the convective boundary conditions to obtain the updated PCE coefficient.

[0033] The iterative formula for the thermal field FDTD method is derived by combining PCE. The randomness of PEC can be realized in the FDTD computational domain. For example, the uncertain variables in this embodiment can be represented by formulas (6), (7), and (8): (6) (7) (8) in, , x The range of =1, 2, 3 is [-1, 1]; This represents the average thermal conductivity. The standard deviation of thermal conductivity; This represents the average specific heat capacity. This represents the standard deviation of specific heat capacity; This represents the average density of the material. The standard deviation of material density.

[0034] Substituting formulas (3)(6)(7)(8) into formula (2) yields: (9) in, Indicates the time step; It represents a polynomial.

[0035] Then, the Galerkin method is used to take the inner product of the test functions. The polynomial basis functions Ψ(ξ) satisfy the orthogonality relation defined by the following equation: (10) in, It is the Kronecker increment function, when ,when And using the orthogonality condition in formula (10), formula (9) is transformed into formula (11). By Projecting onto both sides of formula (9), the Tth order was calculated. l Coefficients: (11) The electromagnetic power dissipation density Q is calculated based on the electric field distribution E. The relationship between Q and E is expressed by formula (12): (12) in, This indicates electrical conductivity.

[0036] By incorporating it as a heat source into the iterative formula for thermal analysis (11), a unidirectional coupling from electromagnetic to thermal processes is achieved: (13) in, m Indicates the first m There are 1 coefficient, the size of which does not exceed 1. P .

[0037] To simulate heat transfer at the boundary of the air model, convective boundary conditions were added to the thermal analysis: (14) in, Indicates the external temperature; h This represents the convective heat transfer coefficient.

[0038] To ensure the validity of the simulation, both electromagnetic and thermal analyses must satisfy the Courant-Friedrichs-Lewis (CFL) stability condition. The electromagnetic field stability condition is given as follows: (15) In the formula, c is the speed of light in free space. Since the parameters in (15) remain unchanged, the stability condition of the PCE-FDTD method is the same as that of the traditional FDTD method. The stability condition used to solve the thermal analysis is: (16) The PCE coefficients are calculated using the FDTD time step at each time step and grid cell, and then these coefficients are used to reconstruct the field stochastic process. In the domain region where the uncertain parameters reside, updating the PCE coefficients requires evaluating the projection in the manner of Equation (13). Although the uncertainties in the material parameters may be confined to specific regions, the uncertainties in the field are assumed to be all regions that the field can reach from the source of the parameter uncertainty. In the region where there are no uncertain parameters, the orthogonal projection decouples the update equations, and each coefficient is updated independently.

[0039] S4. Calculate the mean and variance of the output field based on the updated PCE coefficients.

[0040] Finally, the statistical information of the field is calculated from the PCE solution; for example, the mean and variance of the field components are given by the following formula: (17) (18) in, Represents the coefficient of the lowest-order temperature expansion term; Represents the square of a polynomial expansion; μ This represents the mean; Indicates variance; P Indicates the number of terms expanded; A temperature function representing the effect of variables; Indicates the first l Coefficients of the temperature expansion term.

[0041] Example 2 This embodiment selects a representative case to illustrate the effect of the method of the present invention: Using output statistical moments (mean and standard deviation) as evaluation metrics, the PCE-FDTD algorithm is compared with the traditional MC-FDTD algorithm to confirm the effectiveness, accuracy, and efficiency of the algorithms. The PCE-FDTD results are comprehensively benchmarked against MC-FDTD simulation results to quantitatively assess their accuracy and verify the accuracy of the PCE-FDTD method. Using the time of a single complete uncertainty quantification process as a benchmark, the computational efficiency of the two methods is compared, intuitively revealing the advantages of PCE-FDTD in reducing complexity and verifying the efficiency of the proposed method.

[0042] This embodiment establishes a simple cube model, which is a silicon-based rectangular block with the following dimensions in the x, y, and z directions: Lx =0.5m Ly =0.5m and Lz =0.5m. During the simulation, Dirichlet boundary conditions were used at the interface between the model and the air. The initial temperature of the model was... The ambient temperature is 0K. The silicon-based rectangular block reference model in this embodiment is as follows: Figure 1 As shown, the detailed thermal parameters of the model material are shown in Table 1.

[0043] Table 1 To more intuitively compare the simulation results of the two methods, four sampling points were set up in this simulation, with the coordinates of the sampling points as follows: r 1 (0.25, 0.25, 0.3) r 2 (0.3, 0.15, 0.1) r 3 (0.23, 0.26, 0.05) and r 4 (0.04, 0.26, 0.14), where the mean and variance of T are obtained by repeating the MC-FDTD experiment 1000 times, and the random input adopts a normal distribution. Figure 2 Showing the simulation time t Three-dimensional distribution of transient temperature at 160s. Figure 3 The graph shows the mean temperature change over time at the sampling point under two different methods. A comparison of the temperature distribution curves shows that the temperature change curve obtained using the PCE-FDTD method matches well with the temperature change curve obtained using the analytical solution based on MC-FDTD for calculating the mean temperature. Figure 4 The diagram shows the variance of temperature at the sampling point over time under two different methods. For the calculated temperature variance, when the simulation time is less than 200 s, MC-FDTD and PCE-FDTD (D=1, D=2) coincide, indicating that low-order polynomial chaotic expansion can accurately capture temperature uncertainty within this time period. When the simulation time is greater than 200 s, the D=1 expansion result is slightly lower than MC-FDTD, but the deviation is still within an acceptable range for engineering applications; the D=2 expansion result is consistent with MC-FDTD.

[0044] Under the same hardware platform and parallel configuration, completing 1000 independent samples of MC-FDTD statistics takes 3.43 hours; while PCE-FDTD requires only one deterministic cascade solution: approximately 78 seconds for D=1 and approximately 10 minutes for D=2. This represents a 158-fold and 20-fold improvement in computational efficiency, respectively, significantly reducing the time and space overhead of uncertainty quantification. This verifies the high efficiency of this method in stochastic electromagnetic-thermal coupling problems. Furthermore, the simulation demonstrates that the PCE-FDTD method can effectively perform statistical analysis of thermal fields.

[0045] Example 3 This embodiment also selects a representative case to illustrate the effect of the method of the present invention: This embodiment considers the influence of randomness in GaAs thermal parameters on the thermal field temperature T in the microchip model. The initial ambient temperature is set to 300°C. K All boundaries use convective boundary conditions. x , y , z The convection coefficient in all directions is set to 5. w / ( m²·K Set the horizontal grid size to 2.5 × 10⁻⁷. m The vertical grid size is set to 5×10⁻⁸ m The total number of grid cells is 250,880. To satisfy the CFL stability condition, the simulation time step for the FDTD method is set to 1.4 × 10⁻¹⁶ s. The microchip model structure dimensions are as follows: Figure 5 As shown in Table 2, the detailed electromagnetic and thermal parameters of the model material are presented. In this embodiment, the thermal parameters of GaAs are randomized by 10%, with the three parameters changing independently. Randomized inputs are used during the simulation to explore the statistical characteristics of T.

[0046] Table 2 To more intuitively compare the simulation results of the two methods, the sampling point in this simulation is set at (2.5, 1.5, 5). μm The mean and variance of T were obtained by repeating the MC-FDTD experiment 1000 times, with the random input adopting a normal distribution. Figure 6 The graph shows the mean temperature change over time at the sampling point under two different methods. A comparison of the temperature distribution curves shows that the temperature change curve obtained using the PCE-FDTD method matches well with the temperature change curve obtained using the analytical solution based on MC-FDTD for calculating the mean temperature. Figure 7The diagram shows the variance of temperature at the sampling point over time under two different methods. For the calculated temperature variance, the variances obtained from MC-FDTD and PCE-FDTD (D=1, D=2) almost completely overlap, indicating that low-order polynomial chaotic expansion can accurately capture temperature uncertainty within this time period; when the simulation time is greater than 9 × 10... -13 At time s, the expansion result of D=1 is slightly lower than that of MC-FDTD, and the expansion result of D=2 is slightly higher than that of D=1. Overall, they are quite consistent.

[0047] Under identical high-performance computing platform conditions, performing MC-FDTD statistics on 1000 independent random samples took 81 hours. The example used the PCE-FDTD method proposed in this invention, which can obtain the full-order statistical moments through a single electromagnetic-thermal coupling deterministic cascade solution: the D=1st order polynomial chaotic expansion took about 9 minutes; the D=2nd order expansion took about 21 minutes. Thus, PCE-FDTD improves the uncertainty quantification efficiency by about 540 times (D=1) and 230 times (D=2) respectively, with negligible accuracy loss, significantly reducing the time cost and computational resource overhead of engineering-grade electromagnetic-thermal coupling reliability assessment.

[0048] The embodiments described above are merely preferred embodiments of the present invention and are not intended to limit the scope of the present invention. Various modifications and improvements made to the technical solutions of the present invention by those skilled in the art without departing from the spirit of the present invention should fall within the protection scope defined by the claims of the present invention.

Claims

1. A method for quantitative analysis of uncertainties in the electromagnetic-thermal multiphysics field of a chip based on PCE-FDTD, characterized in that, Includes the following steps: S1. Based on the uncertain input parameters and their probability distribution in the electromagnetic-thermal coupling system, construct an electromagnetic-thermal coupling simulation model containing random parameters, and determine the basis functions and total number of terms of the polynomial chaotic expansion PCE. S2. Based on the electromagnetic-thermal coupling simulation model, the heat conduction equation is discretized using the finite-difference time-domain (FDTD) method, and the random thermal field quantity is expanded using the basis functions of the PCE to obtain the PCE representation of the thermal field quantity. S3. Based on the PCE representation of the thermal field and the PCE representation of the random input parameters, the updated equation of the PCE coefficient is obtained through Galerkin projection, and then solved by combining the electromagnetic power dissipation density and the convective boundary conditions to obtain the updated PCE coefficient. S4. Calculate the mean and variance of the output field based on the updated PCE coefficients.

2. The method for quantifying uncertainties in chip electromagnetic-thermal multiphysics fields based on PCE-FDTD according to claim 1, characterized in that, In step S1, the steps of constructing an electromagnetic-thermal coupling simulation model containing random parameters include: defining the thermal conductivity and specific heat capacity of the medium as random input parameters, and specifying a normal distribution or uniform distribution for each random parameter.

3. The method for quantifying uncertainties in chip electromagnetic-thermal multiphysics fields based on PCE-FDTD according to claim 1, characterized in that, In step S1, the step of determining the basis functions of the polynomial chaotic expansion PCE includes: selecting the corresponding orthogonal polynomial basis according to the distribution type of the standard random variable, wherein Hermite polynomial is selected when the input is a Gaussian distribution and Legendre polynomial is selected when the input is a uniform distribution.

4. The method for quantifying uncertainties in chip electromagnetic-thermal multiphysics fields based on PCE-FDTD according to claim 1, characterized in that, In step S2, the step of obtaining the PCE representation of the thermal field includes: in, i , j , k express x, y, z direction; n Indicates a time step; Indicates the time step; , , Indicates the spatial step size; Using orthogonal polynomial basis functions Unfold the stochastic thermal field for each Yee unit and time step: in, Indicates the first l Coefficients of the temperature expansion term; P Indicates the number of terms expanded; Represents a variable.

5. The method for quantifying uncertainties in chip electromagnetic-thermal multiphysics fields based on PCE-FDTD according to claim 1, characterized in that, In step S3, the method for solving the problem by combining the electromagnetic power dissipation density includes: calculating the electromagnetic power dissipation density Q based on the electric field distribution E, and the relationship between Q and E is as follows: in, This indicates electrical conductivity.

6. The chip electromagnetic-thermal multiphysics uncertainty quantification analysis method based on PCE-FDTD according to claim 5, characterized in that, In step S4, the mean and variance of the output field are calculated based on the updated PCE coefficients. The steps include: in, Represents the coefficient of the lowest-order temperature expansion term; Represents the square of a polynomial expansion; μ This represents the mean; Indicates variance; P Indicates the number of terms expanded; A temperature function representing the effect of variables; Indicates the first l Coefficients of the temperature expansion term.

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