Multi-scale integrated device thermal field analysis method and system based on non-uniform grid local time stepping

Abstract

The invention discloses a multi-scale integrated device thermal field analysis method and system based on non-uniform grid local time stepping, and the method comprises the steps: building an integrated device thermal field solving model, carrying out the discretization of the model through a non-uniform grid, and obtaining the structure information of the model, including the node information and unit information on the grid; the method comprises the following steps of: expanding a thermal field by adopting Taylor series on a non-uniformly subdivided grid, firstly, performing thermal field expansion iteration on a grid with a minimum time step, and calculating thermal field values of different time steps required by iteration by using interpolation at a boundary part of different time steps; then thermal field expansion iteration is carried out on the time step small grids, and so on, thermal field iteration on all the grids in the space is completed, and a thermal field value is obtained; and extracting physical parameters of the integrated device according to the calculated field value. According to the method, the calculation efficiency is remarkably improved, the calculation cost is reduced, and a higher precision level is achieved.

Description

Technical Field

[0001] The present invention relates to a thermal field analysis method for multi-scale integrated devices, and specifically to a multi-scale integrated device thermal field analysis method and system based on non-uniform grid local time stepping. Background Art

[0002] Thermal field analysis has wide applications in integrated devices and complex engineering systems. For such multi-scale problems, traditional time-domain difference methods need to use fine grids to model fine structures, and are limited by the numerical stability condition, so extremely small time steps have to be selected, which greatly reduces the calculation efficiency. The literature E.L. Tan and D.Y. Heh,

[0003] "Stability Analyses of Nonuniform Time-Step LOD-FDTD Methods for Electromagnetic and Thermal Simulations," in IEEE Journal on Multiscale and Multiphysics Computational Techniques, vol. 2, pp. 183-193, 2017 used a local one-dimensional time-domain finite difference method to analyze electromagnetic-thermal coupling simulations, but was limited by the second-order accuracy of the time-domain finite difference method resulting in insufficient accuracy, and only stayed in the first-order problem. To overcome this limitation, the present invention proposes a multi-scale thermal field analysis scheme based on non-uniform grid local time stepping. Summary of the Invention

[0004] The purpose of the present invention is to provide a multi-scale integrated device thermal field analysis method and system based on non-uniform grid local time stepping, which allows different grid regions to perform iterative calculations using corresponding time steps, effectively improving the solution efficiency and obtaining higher accuracy.

[0005] The technical solution for achieving the purpose of the present invention is as follows:

[0006] A multi-scale integrated device thermal field analysis method based on non-uniform grid local time stepping, comprising the steps of:

[0007] Step 1, establish a thermal field solution model for the integrated device, discretize the model using a non-uniform grid to obtain the structural information of the model, including node information and element information on the grid;

[0008] Step 2, on the non-uniformly meshed grid, expand the thermal field using the Taylor series. First, perform the thermal field expansion iteration on the grid with the smallest time step. At the junction of different time steps, use interpolation to calculate the thermal field values at different time steps required for the iteration. Subsequently, perform the thermal field expansion iteration on the grid with the second smallest time step, and so on, to complete the thermal field iteration on all grids within the entire space and obtain the thermal field values.

[0009] Step 3, extract the physical parameters of the integrated device based on the calculated field values.

[0010] Further, the expansion of the thermal field using the Taylor series specifically includes:

[0011] Starting from the heat conduction differential equation, perform discretization to obtain the time-domain recurrence formula as:

[0012]

[0013] where ρ m represents the density of the object; c m represents the specific heat capacity of the object, T(x, y, z, t) represents the transient temperature at this point at time t, t represents time, P d represents the power density of the heat source, D t is the diffusion factor, is the Laplace operator;

[0014] Simplify Equation (1) to obtain:

[0015]

[0016] where the parameter k t = ρ m c m ,

[0017] Take the derivative of Equation (2) with respect to time to obtain And so on to all orders of the derivative of T with respect to time t. According to the Taylor series, we get:

[0018]

[0019] where t represents the time instance, Δt represents the time step used for iteration, M is the set accuracy order, and l is the accumulation variable. On the same grid, according to Equation (3), obtain T at any time.

[0020] Further, the diffusion factor is:

[0021]

[0022] where t represents the time instance, Δt represents the time step used for iteration.

[0023] Further, at the intersection of different time steps, the specific thermal field values of different time steps required for iteration are calculated by interpolation as follows: The thermal field required for iteration at the interface of different time steps is calculated by the interpolation method. When the thermal field of other time steps is required during time iteration, interpolation is performed using the thermal field of the previous time.

[0024] Further, the required thermal field is calculated by the interpolation method as follows:

[0025]

[0026] where n represents the nΔt moment, and Δt f is the fine grid iteration time step, and T c (t) is the thermal field of the coarse grid at time t, and T f is the thermal field of the fine grid. M is the set accuracy order, and l and p are cumulative variables. Through this

[0027] formula, the following is obtained

[0028] A multi-scale integrated device thermal field analysis system based on non-uniform grid local time stepping includes:

[0029] An integrated device thermal field solution model construction unit for establishing an integrated device thermal field solution model, discretizing the model using non-uniform grids to obtain the structural information of the model, including node information and element information on the grids;

[0030] A field value solution unit that expands the thermal field using Taylor series on the non-uniformly partitioned grids. First, the thermal field expansion iteration is performed on the grids with the smallest time step. At the intersection of different time steps, the thermal field values of different time steps required for iteration are calculated by interpolation; then, the thermal field expansion iteration is performed on the grids with the second smallest time step, and so on, to complete the thermal field iteration on all grids in the entire space and obtain the thermal field values;

[0031] A physical parameter extraction unit that extracts the physical parameters of the integrated device according to the calculated field values.

[0032] A multi-scale integrated device thermal field analysis device includes: a memory, a processor, and a computer program stored on the memory. When the processor executes the computer program, the steps of the multi-scale integrated device thermal field analysis method are implemented.

[0033] A computer storage medium stores an executable program, and when the executable program is executed by a processor, the steps of the multi-scale integrated device thermal field analysis method are implemented.

[0034] Compared with the prior art, the present invention has the following remarkable advantages: (1) Based on the Taylor series expansion and through the high-order derivative reconstruction technology, the present invention can realize the iteration of the thermal field on grids with different time steps, accurately describe the non-uniform grid thermal field distribution, and obtain higher accuracy than the traditional time-domain difference method; (2) In multi-scale thermal analysis, different grid regions are allowed to adopt different time steps, which can magnify the time. The step size significantly reduces the computational complexity, reduces the time required for calculation, and improves the numerical stability; (3) The flexible iteration strategy of the present invention is particularly suitable for thermal field problems containing multiple characteristic scales, showing important engineering application value. BRIEF DESCRIPTION OF THE DRAWINGS

[0035] Figure 1 It is a schematic diagram of the iteration of different time steps in the computational domain.

[0036] Figure 2 It is a schematic diagram of the chip heat dissipation model.

[0037] Figure 3 It is a schematic diagram of the composition structure of the chip heat dissipation model.

[0038] Figure 4 It is a temperature distribution diagram along the X direction at the observation line (Y = 5 mm, Z = 5 mm). DETAILED DESCRIPTION OF THE EMBODIMENTS

[0039] The present invention will be further described in detail below with reference to the accompanying drawings.

[0040] The present invention provides a multi-scale integrated device thermal field analysis method based on non-uniform grid local time stepping, and the steps are as follows:

[0041] In the first step, a solution model is established, and the model is discretized using non-uniform grids to obtain the structural information of the model, including node information and element information on the grids;

[0042] In the second step, on the non-uniformly meshed grids, the thermal field is expanded using the Taylor series. First, the thermal field iteration is performed on the grids with the smallest time step. At the intersection of different time steps, interpolation is used to calculate the thermal field values of different time steps required for the iteration; then the thermal field iteration is performed on the grids with the second smallest time step; and so on, until the thermal field iteration on all the grids in the entire space is completed.

[0043] This method expands the thermal field through the Taylor series, performs thermal field iteration with the same time step on the same grids, and realizes data transfer between different time step regions through interpolation, so as to realize the complete thermal field iteration in the entire space and time.

[0044] The thermal field expanded by the Taylor series can realize iteration on grids of the same size;

[0045] Starting from the heat conduction differential equation, discretize it, and then obtain the time-domain recurrence formula.

[0046]

[0047] Among them, ρ m represents the density of the object, with the unit of (kg / m 3 ); c m represents the specific heat capacity of the object, with the unit of (J / (kg·℃)); T(x, y, z, t) represents the transient temperature at this point at time t, with the unit of (K); t represents time; P d represents the power density of the heat source, with the unit of (W / m 3 ); the proportionality factor D t is the diffusion factor, and the expression is:

[0048]

[0049] Simplify equation (1) to obtain:

[0050]

[0051] Among them, k t = ρ m c m . Take the derivative of equation (3) with respect to time to obtain By analogy, the derivatives of T with respect to time t of all orders can be obtained. According to the expansion of the Taylor series:

[0052]

[0053] Expand to obtain:

[0054]

[0055] Among them, t represents the time instant, and Δt represents the time step used for iteration. On the same grid, according to equation (5), T at any time can be obtained, and thus the heat field at any time can be obtained.

[0056] Based on the heat field expanded by the above Taylor series, the specific implementation method of step 2 is as follows:

[0057] Combined with, for example Figure 1 , assume that the large time step is Δt = dt, and the small time step is Δt = dt / 5. Iterate with the large time step on the coarse grid and iterate with the small time step on the fine grid.

[0058] (a) At the starting moment, t = (n + 0.2)·Δt. Let i0 be the position of the fine grid. When i < i0, it is the fine grid, and when i > i0, it is the coarse grid. When i0 is not at the boundary between the coarse and fine grids, (Here, Δt = dt) to obtain Similarly (Here, Δt = dt) to obtain Then, the following are obtained successively etc.

[0059] When i0 is the boundary between the fine and coarse grids, the coarse grid value at the previous moment is required when iterating the thermal field value at i0. At this time, t = nΔt, which is the initial value of the small time step.

[0060] (b) At time t = (n + 0.4)·Δt, update through iteration to obtain When i0 is not at the boundary between the fine and coarse grids (Here, Δt = dt) to obtain Similarly (Here, Δt = dt) to obtain Then, the following are obtained successively etc. When i0 is the boundary between the fine and coarse grids, the coarse grid value at the previous moment is required when iterating the thermal field value at i0. At this time, interpolation is required:

[0061] After obtaining the coarse grid value, update the value of i0 at the junction:

[0062]

[0063] Then, iterate the thermal field at time t = (n + 0.6)·Δt, and so on.

[0064] Third step, data post - processing, extract relevant physical parameters according to the calculated field values.

[0065] This embodiment also provides a multi - scale integrated device thermal field analysis system based on non - uniform grid local time stepping, including:

[0066] An integrated device thermal field solution model construction unit, used to establish an integrated device thermal field solution model, discretize the model using non - uniform grids, and obtain the structural information of the model, including node information and element information on the grids;

[0067] A field value solution unit, on the non - uniformly divided grids, expand the thermal field using Taylor series. First, perform thermal field expansion iteration on the grids with the smallest time step. At the junction of different time step lengths, use interpolation to calculate the thermal field values of different time step lengths required for iteration; then perform thermal field expansion iteration on the grids with the second - smallest time step, and so on, to complete the thermal field iteration on all grids in the entire space and obtain the thermal field values;

[0068] A physical parameter extraction unit, according to the calculated field values, extract the physical parameters of the integrated device.

[0069] This embodiment also provides a thermal field analysis device for multi-scale integrated devices, including: a memory, a processor, and a computer program stored on the memory. When the processor executes the computer program, the steps of the multi-scale integrated device thermal field analysis method are implemented.

[0070] This embodiment also provides a computer storage medium, which stores an executable program. When the executable program is executed by a processor, the steps of the multi-scale integrated device thermal field analysis method are implemented.

[0071] It should be noted that steps 1 and 3 in the method of the present invention are conventional technologies in the art and will not be described in detail herein.

[0072] To verify the correctness and effectiveness of the present invention, the thermal characteristics of a chip heat dissipation model are analyzed below.

[0073] The LF-FDTD and ADER-LTS-FDTD algorithms are respectively used to perform thermal simulation on the following Figure 2 shown chip heat dissipation model. The size of this model is 6.82mm×10mm×10mm, and its composition structure is as Figure 3 shown. Among them, the bottom medium is the bottom plate with a thickness of 0.3mm, the wafer is a silicon wafer with a thickness of 0.5mm, the TIM is a thermal conductive medium with a thickness of 0.02mm, the heat sink is a radiator with a thickness of 1mm, the height of the heat dissipation fins is 5mm, the thickness is 0.5mm, and the rest is air. The material properties of each component are shown in Table 3.3.1. Die is the heat source with a power density of 1×10 10 W / m 3 . The coarse grid meshing size is Δx = 0.1mm, Δy = 0.25mm, Δz = 0.5mm. The distribution of the fine grid in the x direction is 0.8mm - 0.84mm. The fine grid meshing size in the x direction is Δx = 0.02mm, and the meshing size ratio is 5:1. The ratio of the large and small time steps is 18:1. As Figure 3 shown, the bottom of the model is at a fixed temperature of 27°C, and the rest of the boundary conditions are thermally insulated.

[0074] As Figure 4 , the overall computational amount ratio of the LF-FDTD method to ADER-LTS-FDTD is 18.2:1, and the ratio of the time used is 15.7:1. Considering the additional pre-processing and operations required during program implementation, the obtained time ratio and computational amount ratio can be considered approximately equal. In addition, the temperature distributions obtained by the two methods are approximately equal and are similar to the results of the commercial software COMSOL. The computational efficiency is increased by 1.68 times compared to COMSOL. Therefore, we can consider that the ADER-LTS method still greatly improves the computational efficiency while ensuring the accuracy when calculating the thermal field.

[0075] The method of the present invention is based on the Taylor series expansion as the mathematical basis. Through the high-order derivative reconstruction technology, it realizes the accurate description of the thermal field distribution of non-uniform grids. It allows different grid regions to adopt different time steps, significantly reducing the computational complexity and improving the numerical stability. Compared with the traditional finite-difference time-domain method, this method not only significantly improves the computational efficiency, reduces the computational cost, but also reaches a higher accuracy level, and the implementation method is simple, having strong practical engineering application value in the field of integrated device simulation and thermal effect analysis.

[0076] Obviously, those skilled in the art can make various changes and modifications to the embodiments of the present invention without departing from the spirit and scope of the embodiments of the present invention. Thus, if these modifications and variations of the embodiments of the present invention fall within the scope of the claims of the present invention and their equivalent technologies, the present invention also intends to include these changes and modifications.

Claims

1. A multi-scale integrated device thermal field analysis method based on non-uniform grid local time stepping, characterized in that: Includes steps: Step 1, establish a thermal field solution model for the integrated device, discretize the model using a non-uniform grid, and obtain the structural information of the model, including node information and unit information on the grid; Step 2: On the non-uniformly divided grid, the thermal field is expanded using Taylor series. First, the thermal field iteration is performed on the grid with the smallest time step. At the boundary of different time steps, the thermal field values ​​of different time steps required for iteration are calculated using interpolation. Then, the thermal field expansion iteration is performed on the grid with the smallest time step. This process is repeated to complete the thermal field iteration on all grids in the space and obtain the thermal field value. Step 3: Extract the physical parameters of the integrated device based on the calculated field values.

2. The multi-scale integrated device thermal field analysis method based on non-uniform grid local time stepping according to claim 1 is characterized in that: The use of Taylor series to expand the thermal field specifically includes: Starting from the heat conduction differential equation, the time domain recursive formula is obtained by discretization: Among them, ρ m Indicates the density of the object; c m represents the specific heat capacity of the object, T(x, y, z, t) represents the transient temperature at the point t, t represents the time, P d Denotes the power density of the heat source, D t is the diffusion factor, is the Laplace operator; Simplifying formula (1) we can get: Among them, the parameter k t =ρ m c m ; Taking the derivative of formula (2) with respect to time, we get By analogy, the derivatives of T with respect to time t can be obtained according to Taylor series: Among them, t represents the time, Δt represents the time step adopted in the iteration, M is the set accuracy order, and l is the accumulated variable.

3. The multi-scale integrated device thermal field analysis method based on non-uniform grid local time stepping according to claim 2 is characterized in that: The diffusion factor is: Among them, t represents the time instant, and Δt represents the time step adopted in the iteration.

4. The multi-scale integrated device thermal field analysis method based on non-uniform grid local time stepping according to claim 1 is characterized in that: At the boundary of different time steps, interpolation is used to calculate the thermal field values ​​of different time steps required for iteration. Specifically: the thermal field required for iteration at the interface of different time steps is calculated by the interpolation method. When the thermal field of other time steps is needed during time iteration, the thermal field of the previous time is interpolated.

5. The multi-scale integrated device thermal field analysis method based on non-uniform grid local time stepping according to claim 4 is characterized in that: The required thermal field is calculated by interpolation method as follows: Where n represents the nΔtth moment, Δt f is the fine grid iteration time step, T c (t) is the thermal field of the coarse grid at time t, T f is the thermal field of the fine grid, p is the cumulative variable, and the formula is 6. A multi-scale integrated device thermal field analysis system for implementing the method described in any one of claims 1 to 5, characterized in that: include: The integrated device thermal field solution model building unit is used to establish the integrated device thermal field solution model, discretize the model using a non-uniform grid, and obtain the structural information of the model, including node information and unit information on the grid; The field value solving unit uses Taylor series to expand the thermal field on the non-uniformly divided grid. First, the thermal field expansion iteration is performed on the grid with the smallest time step. At the junction of different time steps, the thermal field values ​​of different time steps required for iteration are calculated by interpolation. Then, the thermal field expansion iteration is performed on the grid with the smallest time step, and so on. The thermal field iteration on all the grids in the space is completed to obtain the thermal field value. The physical parameter extraction unit extracts the physical parameters of the integrated device according to the calculated field values.

7. A multi-scale integrated device thermal field analysis device, characterized in that: include: A memory, a processor and a computer program stored in the memory, wherein when the processor executes the computer program, the steps of the multi-scale integrated device thermal field analysis method according to any one of claims 1 to 5 are implemented.

8. A computer storage medium, characterized in that: The computer storage medium stores an executable program, and the executable program is executed by a processor to implement the steps of the multi-scale integrated device thermal field analysis method according to any one of claims 1 to 5.

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