Chip thermal analysis method and system based on fourier series domain decomposition

Abstract

The application provides a chip thermal analysis method and system based on Fourier series region decomposition, and the method comprises the following steps: obtaining a chip region distribution data set, a power consumption distribution data set and a verification data set; calculating the temperature of each region of the chip; calculating the solution u n (y) is brought into a preset expression to perform a loop operation, so that function value traversal and accumulation operation are completed, and a temperature distribution result T(x, y) is obtained; the temperature distribution result T(x, y) is compared with a finite element simulation result, the accuracy of the temperature distribution result T(x, y) is judged, the numerical integration method and the piecewise analytical method are compared, and an algorithm running accuracy and speed judgment result is obtained. The application solves the technical problems of low thermal analysis rate and accuracy and low temperature solving efficiency.

Description

Technical Field

[0001] This invention relates to the field of chip analysis, and more specifically to a chip thermal analysis method and system based on Fourier series region decomposition. Background Technology

[0002] As chip size continues to shrink and power density continues to increase, some of the electrical energy consumed by the circuit is converted into heat, leading to increased internal chip temperature and reduced circuit reliability. Integrating temperature sensors into the chip or using thermal imaging technology to measure its heat distribution faces numerous spatial challenges. Therefore, a technique has been proposed to accurately predict chip heat distribution in the early stages of chip design, allowing for continuous optimization of the chip design based on the prediction results.

[0003] Chip thermal analysis is divided into two categories: numerical methods and simulation methods. Numerical methods use the finite difference method or finite element method to transform the thermal equation into an RC network equation for solution. This process requires meshing the entire chip, and the fineness of the mesh determines the accuracy of the thermal analysis. For example, the existing invention patent application document CN117218316A, entitled "Mesh Space Partitioning Method for Thermal Analysis of Chip Packaging Structure Based on Advanced Process," extracts the structural contour based on the physical model of the chip packaging structure. It then uses a mesh generation algorithm to generate rippled meshes, through meshes, and tree meshes. The rippled meshes are used to meet basic mesh density requirements and describe the contour of the packaging structure. The through meshes are used to describe the contours of complex heat sources and boundary conditions within the structure. The tree meshes are used for adaptive refinement of heat sources and their surroundings. The combination of these three types of meshes achieves a high degree of precision in thermal analysis of the chip's internal structure. The adaptation of complex structures and complex heat source distributions in chips, and the existing invention patent application document CN105718638A entitled "A Three-Dimensional Thermal Analysis Method for Chips," includes the following steps: S1, acquiring chip data and generating a transfer region table according to a general random algorithm; S2, calculating the transfer probabilities required for general random walks based on the chip data; S3, inputting the coordinates of the temperature point to be calculated within the heat source region, and combining the corresponding region in the chip thermal analysis model based on a hybrid random walk algorithm, performing temperature calculations using the general random walk method based on the transfer probabilities, and performing temperature calculations using the suspended random walk method based on the transfer region table; S4, after completing the temperature calculation for the coordinates, returning to step S3 to perform the temperature calculation for the next temperature point to be calculated, until the temperature calculations for all input coordinate points are completed. Furthermore, many studies based on numerical methods focus on reducing computation time. This method has the ability to flexibly handle complex structures and is mainly applied to temperature analysis in the back-end design stage, such as later layout stages. Early-stage thermal analysis in chip design needs to provide reasonable and accurate temperature predictions with a relatively small computational load. Analytical methods can avoid dividing the entire chip substrate into a network and flexibly obtain the temperature distribution of a specified region. For example, analytical methods based on Green's function have greatly improved the speed of chip thermal analysis. However, the above techniques have the following drawbacks:

[0004] First, although the finite element method provides high accuracy, it requires meshing the entire chip substrate, resulting in a huge amount of computation and a long computation time.

[0005] Second, traditional analytical methods such as the Green function have significantly improved computation speed, but the Green function of a chip itself is very complex.

[0006] In summary, existing technologies suffer from low thermal analysis rates and accuracy, as well as low temperature solution efficiency. Summary of the Invention

[0007] The technical problem to be solved by this invention is: how to solve the problems of low thermal analysis rate and accuracy, and low temperature solution efficiency in the prior art.

[0008] This invention solves the above-mentioned technical problems by employing the following technical solution: a chip thermal analysis method based on Fourier series region decomposition, comprising:

[0009] S1. Obtain the chip region distribution dataset, power consumption distribution dataset, and verification dataset;

[0010] S2. Based on the chip region dataset, divide the rectangular chip of the predetermined area into at least two chip regions. Process the data to obtain the position coordinates and power consumption of each chip region, and then process the data to obtain the chip temperature distribution. Use the method of separation of variables to expand and transform the non-homogeneous differential equation, and then transform and solve the homogeneous differential equation to obtain the solution u. n (y) is used to process and obtain the temperature of each region of the chip;

[0011] S3, solve u n (y) is substituted into the preset expression for loop operation to complete the function value traversal and accumulation operation, and obtain the temperature distribution result T(x,y);

[0012] S4. Compare the temperature distribution result T(x,y) with the finite element simulation result to determine the accuracy of the temperature distribution result T(x,y). Compare the numerical integration method and the analytical method to obtain the algorithm running accuracy and speed judgment result.

[0013] This invention employs an analytical method, using variable substitution to calculate and realize the theoretical general solution of the partial differential heat transfer equation, thereby accelerating the rate of thermal analysis.

[0014] In a more specific technical solution, step S2 includes:

[0015] S21. Divide the rectangular chip of the preset area into no less than 2 chip regions according to the chip region dataset, and process to obtain the position coordinates and power consumption of each chip region.

[0016] S22. Set the equations and boundary conditions for the relationship between power consumption and temperature, and process them to obtain the chip temperature distribution.

[0017] S23. Expand f(x,y) using the method of separation of variables and substitute it into the equation relating power consumption and temperature to obtain a non-homogeneous differential equation.

[0018] S24. Perform variable substitution on the non-homogeneous differential equation to obtain a homogeneous differential equation. Solve the homogeneous equation to obtain the solution u. n (y);

[0019] S25. Let the function f n (y) There are num_seg_f segments. Based on these segments, we can obtain num_seg_f different boundary coefficients c1 and c2, and thus obtain 2*(num_seg_f-1) boundary conditions.

[0020] S26. Based on 2*(num_seg_f-1) boundary conditions, and the different constant coefficients c1 and c2 of the corresponding num_seg_f groups, set the coefficient matrix A and the boundary coefficient matrix c of the system of equations, and construct the matrix equation system accordingly.

[0021] S27. Solve the system of matrix equations to obtain the solution u. n (y), based on which the generalized equation is obtained.

[0022] This invention employs an analytical method to derive the theoretical general solution of the partial differential heat transfer equation through variable substitution, and then implements this general solution in code, which greatly accelerates the speed of thermal analysis.

[0023] In a more specific technical solution, in step S24, the boundary parameters of different chip regions are substituted into a preset function relationship using the following logic to obtain u. n (y) and function f n (y).

[0024] In a more specific technical solution, u is obtained by using the following logic. n (y) and function f n (y):

[0025]

[0026]

[0027] In a more specific technical solution, step S25 includes:

[0028] S251. Substituting different boundary conditions, in the function f n For each boundary point of (y), the relationship between the boundary point parameters is obtained;

[0029] S252. Based on 2*(num_seg_f-1) boundary conditions, determine the boundary parameter relationships on the boundary.

[0030] In a more specific technical solution, step S251 uses the following logic to express the relationship between the boundary point parameters:

[0031] u n (y - )=u n (y + )

[0032]

[0033] In a more specific technical solution, step S252 uses the following logic to express the boundary parameter relationship:

[0034]

[0035]

[0036] In a more specific technical solution, step S26 uses the following logical expression for the matrix equations:

[0037] Ac = b.

[0038] In a more specific technical solution, step S27 uses the following logical expression to represent the general equation:

[0039]

[0040] In more specific technical solutions, chip thermal analysis systems based on Fourier series region decomposition include:

[0041] The data module is used to obtain chip region distribution datasets, power consumption distribution datasets, and verification datasets.

[0042] The temperature calculation module divides a rectangular chip of a predetermined area into at least two chip regions based on the chip region dataset. It then processes the position coordinates and power consumption of each chip region to obtain the chip temperature distribution. Using the separation of variables method, it expands and transforms the non-homogeneous differential equation to obtain and solve the homogeneous differential equation, ultimately yielding the solution u. n (y) is used to process and obtain the temperature of each region of the chip. The temperature solution module is connected to the data module.

[0043] The temperature distribution processing module is used to process the solution u n (y) is substituted into the preset expression for loop operation to complete the function value traversal and accumulation operation, and the temperature distribution result T(x,y) is obtained. The temperature distribution processing module is connected to the temperature solution module.

[0044] The algorithm determination module is used to compare the temperature distribution result T(x,y) with the finite element simulation result, determine the accuracy of the temperature distribution result T(x,y), compare the numerical integration method and the piecewise analytical method, and obtain the algorithm running accuracy and speed determination result. The algorithm determination module is connected to the temperature distribution processing module.

[0045] The present invention has the following advantages over the prior art:

[0046] This invention employs an analytical method, using variable substitution to calculate and realize the theoretical general solution of the partial differential heat transfer equation, thereby accelerating the rate of thermal analysis.

[0047] This invention employs an analytical method to derive the theoretical general solution of the partial differential heat transfer equation through variable substitution, and then implements this general solution in code, which greatly accelerates the speed of thermal analysis.

[0048] This invention solves the technical problems of low thermal analysis rate and accuracy, and low temperature solution efficiency in the prior art. Attached Figure Description

[0049] Figure 1 This is a schematic diagram of the basic steps of the chip thermal analysis method based on Fourier series region decomposition according to Embodiment 1 of the present invention.

[0050] Figure 2 This is a schematic diagram illustrating the specific steps involved in calculating the temperature of each region of the chip in Embodiment 1 of the present invention;

[0051] Figure 3 This is a schematic diagram illustrating the specific implementation steps of the chip thermal analysis method based on Fourier series region decomposition in Embodiment 2 of the present invention. Detailed Implementation

[0052] To make the objectives, technical solutions, and advantages of the embodiments of the present invention clearer, the technical solutions of the embodiments of the present invention will be clearly and completely described below in conjunction with the embodiments of the present invention. Obviously, the described embodiments are only some embodiments of the present invention, 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.

[0053] Example 1

[0054] like Figure 1 As shown, the chip thermal analysis method based on Fourier series domain decomposition provided by this invention includes the following basic steps:

[0055] Step S1: Obtain the chip region distribution dataset, power consumption distribution dataset, and verification dataset;

[0056] Step S2: Calculate the temperature of each region of the chip;

[0057] like Figure 2 As shown, in this embodiment, step S2, which calculates the temperature of each region of the chip, further includes the following specific steps:

[0058] Step S21: Divide a rectangular chip with an area of ​​a*b into several different regions and name them according to the chip region dataset;

[0059] In this embodiment, based on the experimental dataset, the chip modules are named: Icache_0, Dcache_0, Bpred_0_0, Bpred_1_0; in this embodiment, each region represents a different module inside the chip, and each different module has a different circuit design, thus producing different power consumption; the power consumption of each module is given by the power consumption distribution dataset.

[0060] In this embodiment, the `genfromtxt` function is used to import the chip module distribution diagram and the power consumption of each module. The starting positions of each module are then selected and sorted to obtain the coordinates of each module and its corresponding power consumption. In the experiment, the y-axis is divided into `num_seg_y` segments, and the x-axis is divided into `num_seg_x` segments.

[0061] Step S22: List the governing equations for power consumption and temperature, and the corresponding boundary conditions:

[0062]

[0063]

[0064]

[0065] Where T(x,y) represents the temperature distribution at different locations within the chip, m is the equivalent coefficient of thermal conduction in the vertical direction of the chip, a represents the width of the chip in the horizontal direction, b represents the length of the chip in the horizontal direction, and f(x,y) is a function determined by the power consumption distribution dataset and the module coordinates.

[0066] Step S23: Based on the boundary conditions, expand the temperature function T(x,y) by separation of variables to obtain:

[0067]

[0068] Step S24: Similarly, expand f(x,y) using the method of separation of variables and substitute it into the original equation. This transforms the partial differential equation into a non-homogeneous differential equation. Then, substitute the variables in the non-homogeneous differential equation to obtain a homogeneous differential equation. Solve the homogeneous equation to obtain:

[0069]

[0070]

[0071] In this embodiment, c1 and c2 under different n values ​​are substituted into the expression to obtain u. n (y);

[0072] Step S25, due to f n (y) is a piecewise function, where different values ​​of f correspond to different coefficients c1 and c2. Let f be a piecewise function.n (y) If there are num_seg_f segments, then there are num_seg_f groups of different constant coefficients c1 and c2, which are used to determine all constant coefficients;

[0073] In this embodiment, different boundary conditions need to be substituted. In f n For each boundary point of (y), we have:

[0074] u n (y - )=u n (y + )

[0075]

[0076] There are a total of 2*(num_seg_f-1) boundary conditions, and the following conditions apply to the boundaries:

[0077]

[0078]

[0079] In this embodiment, different n values ​​correspond to different coefficients c1 and c2. The n values ​​in c) are truncated to n = 40; first determine f n For different values ​​of n, first integrate with respect to x, then fix the values ​​of each segment on the y-axis as y1, y2...y, for a total of num_seg_y, to obtain the piecewise function f with respect to the y-axis for different values ​​of n. n (y); In this embodiment, for different n values, the obtained f is then... n Substituting the (y) value into the matrix equation and solving it, we obtain the final c1 and c2;

[0080] Step S26: From the above, we obtain a total of 2*num_seg_f boundary conditions, corresponding to num_seg_f different constant coefficients c1 and c2. Let the coefficients of the system of equations be set as matrix A, and the constant coefficients c1 and c2 as matrix c, resulting in the following system of equations:

[0081] Ac = b

[0082] In this embodiment, 0-y is divided into num_seg_y segments. For different n values, the resulting f... n Substitute the (y) value into the matrix equation to solve.

[0083] Step S27: Solve this matrix equation to obtain all the constant coefficients c, that is, obtain u. n (y) All solutions yield the governing equation:

[0084]

[0085] Step S3: Implement the theoretical calculation formula derived above using code;

[0086] In this embodiment, u n (y) is substituted into the expression and looped through each obtained value of n. The obtained function values ​​are accumulated to obtain the final temperature distribution result T(x, y).

[0087] Step S4: Compare the results obtained from running the code with the finite element simulation results to determine whether the results are accurate. Also, compare the numerical integration method and the piecewise analytical method to determine the accuracy and speed of the algorithm.

[0088] Example 2

[0089] like Figure 3 As shown in this embodiment, the chip thermal analysis method based on Fourier series domain decomposition further includes the following specific implementation steps:

[0090] S101, Import distributed module and power consumption, set parameter cutoff value;

[0091] S102. Filter and sort the imported module coordinates;

[0092] S103. Fix the y-values ​​on the coordinate axis and integrate over the x-axis;

[0093] S104. Substitute the function obtained after integration into the matrix equation to solve;

[0094] S105. Find the general solution and particular solution of the differential equation and substitute them into the temperature function T(x,y);

[0095] S106. Determine whether the result matches the simulation result. If yes, end the temperature calculation. If no, repeat steps S101 to S105.

[0096] In this embodiment, to verify the performance of the present invention, experiments were conducted on three publicly available datasets (ev6_3D, Chiplet_Core0, and Chiplet_Core0_7), and the analytical method and the algorithm using ordinary numerical integration were analyzed and compared. All experiments were conducted according to the experimental specifications of the corresponding datasets. As can be seen from the comparisons in Tables 1, 2, and 3 below, the analytical method outperforms the algorithm using ordinary numerical integration in both analysis speed and accuracy. This fully demonstrates the important role of analytical algorithms in chip thermal analysis. In the above three datasets, the parameter calculation times of the present invention are 0.09s, 0.06s, and 0.53s, respectively. Compared with currently published mainstream algorithms, the experimental results show higher computational efficiency and accuracy. The analysis speed and result accuracy of the present invention are superior to the mainstream algorithms of the ev6_3D, Chiplet_Core0, and Chiplet_Core0_7 datasets.

[0097] Table 1 Experimental data for the ev6_3D dataset

[0098] Integration methods Result calculation time (s) Numerical methods 1.61 Parsing method 0.09

[0099] Table 2 Experimental data for the Chiplet0_Core0_1 dataset

[0100]

[0101]

[0102] Table 3 Experimental data for the Chiplet0_Core0_7 dataset

[0103] Integration methods Result calculation time (s) Numerical methods 1.70 Parsing method 0.53

[0104] In summary, this invention employs an analytical method to calculate and realize the theoretical general solution of the partial differential heat transfer equation through variable substitution, thereby accelerating the rate of thermal analysis. This invention utilizes an analytical method to calculate the temperature distribution, which can effectively accelerate the algorithm's solution for temperature and improve the accuracy of chip thermal analysis.

[0105] This invention employs an analytical method to derive the theoretical general solution of the partial differential heat transfer equation through variable substitution, and then implements this general solution in code, which greatly accelerates the speed of thermal analysis.

[0106] This invention solves the technical problems of low thermal analysis rate and accuracy, and low temperature solution efficiency in the prior art.

[0107] The above embodiments are only used to illustrate the technical solutions of the present invention, and are not intended to limit it. Although the present invention has been described in detail with reference to the foregoing embodiments, those skilled in the art should understand that modifications can still be made to the technical solutions described in the foregoing embodiments, or equivalent substitutions can be made to some of the technical features. Such modifications or substitutions do not cause the essence of the corresponding technical solutions to deviate from the spirit and scope of the technical solutions of the embodiments of the present invention.

Claims

1. A chip thermal analysis method based on Fourier series domain decomposition, characterized in that, The method includes: S1. Obtain the chip region distribution dataset, power consumption distribution dataset, and verification dataset; S2. Based on the chip region dataset, divide the rectangular chip of the predetermined area into at least two chip regions. Process the data to obtain the position coordinates and power consumption of each chip region, and then process the data to obtain the chip temperature distribution. Use the method of separation of variables to expand and transform the non-homogeneous differential equation, and then transform and solve the homogeneous differential equation to obtain the solution. The temperature of each region of the chip is obtained based on this information. n Indicates the first n Each chip area; S3, the solution The values ​​are substituted into a preset expression and looped through to complete the function value traversal and accumulation, thus obtaining the temperature distribution result. ; S4. The temperature distribution results are... The temperature distribution result is compared with the finite element simulation results to determine the result. The accuracy of the algorithm is compared between numerical integration methods and piecewise analytical methods to obtain the results of the algorithm's running accuracy and speed judgment.

2. The chip thermal analysis method based on Fourier series domain decomposition according to claim 1, characterized in that, Step S2 includes: S21. Divide the rectangular chip of the preset area into no less than two chip regions according to the chip region dataset, and process to obtain the position coordinates and power consumption of each chip region. S22. Set the equations and boundary conditions for the relationship between power consumption and temperature, and process them to obtain the chip temperature distribution. S23, will f(x,y) The equation relating power consumption and temperature is expanded using the method of separation of variables and substituted into it to obtain a non-homogeneous differential equation, where, It is a function determined by the power consumption distribution dataset and module coordinates; S24. Perform variable substitution on the non-homogeneous differential equation to obtain a homogeneous differential equation, and solve the homogeneous equation to obtain the solution. ; S25. Let the function be... f n (y) have num_seg_f Section, based on obtaining num_seg_f Group of different boundary coefficients c 1, c 2, we get 2 (num_seg_f - 1) boundary conditions, where, f n (y) It is a piecewise function; S26, according to the above 2 (num_seg_f - 1) boundary conditions, corresponding to num_seg_f Group of different constant coefficients c 1, c 2. Set the coefficient matrix of the system of equations A Boundary coefficient matrix c Based on this, a system of matrix equations is constructed; S27. Solve the system of matrix equations to obtain the solution. u n (y) Based on this, the generalized equation is obtained.

3. The chip thermal analysis method based on Fourier series domain decomposition according to claim 2, characterized in that, In step S24, the boundary parameters of different chip regions are substituted into a preset function relationship using the following logic to obtain the solution. and functions f n (y) .

4. The chip thermal analysis method based on Fourier series domain decomposition according to claim 3, characterized in that, Using the following logic, the solution is obtained. and functions f n (y) : in, It is the equivalent coefficient of thermal conductivity in the vertical direction of the chip. n Indicates the first n Each chip area, This represents the length of the chip along the horizontal direction.

5. The chip thermal analysis method based on Fourier series domain decomposition according to claim 2, characterized in that, Step S25 includes: S251. Substituting different boundary conditions, in the function... For each boundary point, obtain the boundary point parameter relationship; S252, according to 2 (num_seg_f - 1) boundary conditions, which determine the boundary parameter relationships on the boundary; 6. The chip thermal analysis method based on Fourier series domain decomposition according to claim 5, characterized in that, In step S251, the relationship between the boundary point parameters is expressed using the following logic: 。 7. The chip thermal analysis method based on Fourier series domain decomposition according to claim 5, characterized in that, In step S252, the boundary parameter relationship is expressed using the following logic: 。 8. The chip thermal analysis method based on Fourier series domain decomposition according to claim 2, characterized in that, In step S26, the matrix equations are expressed using the following logic: 。 9. The chip thermal analysis method based on Fourier series domain decomposition according to claim 2, characterized in that, In step S27, the generalized equation is expressed using the following logic: in, It is the equivalent coefficient of thermal conductivity in the vertical direction of the chip, where T represents temperature. It is a function determined by the power distribution dataset and module coordinates.

10. A chip thermal analysis system based on Fourier series domain decomposition, characterized in that, The system includes: The data module is used to obtain chip region distribution datasets, power consumption distribution datasets, and verification datasets. The temperature calculation module is used to divide a rectangular chip of a predetermined area into at least two chip regions based on the chip region dataset. It processes the data to obtain the position coordinates and power consumption of each chip region, and then processes this data to obtain the chip temperature distribution. Using the method of separation of variables, it expands and transforms the data to obtain a non-homogeneous differential equation, and then transforms and solves the homogeneous differential equation to obtain the solution. The temperature of each region of the chip is obtained by processing the data, and the temperature calculation module is connected to the data module. Temperature distribution processing module, used to process the solution The values ​​are substituted into a preset expression and looped through to complete the function value traversal and accumulation, thus obtaining the temperature distribution result. The temperature distribution processing module is connected to the temperature solution module. n Indicates the first n Each chip area; The algorithm determination module is used to determine the temperature distribution results. The temperature distribution result is compared with the finite element simulation results to determine the result. The accuracy of the algorithm is compared with that of the numerical integration method and the piecewise analytical method to obtain the algorithm running accuracy and speed judgment result. The algorithm judgment module is connected to the temperature distribution processing module.

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