Integrated device stress field simulation method and system based on arbitrary high-order multi-scale local time stepping

Abstract

The invention discloses an integrated device stress field simulation method and system based on any high-order multi-scale local time stepping, and the method carries out the Taylor series expansion of a stress field in a non-uniform grid, and achieves the precise description of the distribution of the stress field through a high-order derivative reconstruction technology. According to the method, high-precision time integration is realized by combining an arbitrary order (ADER) technology, and meanwhile, a local time step (LTS) strategy is introduced, so that different grid regions can adopt differentiated time steps, and therefore, the calculation complexity is remarkably reduced, and the numerical stability is improved. Compared with a traditional time domain finite difference method, the method has the advantages that calculation efficiency is greatly improved, calculation resource consumption is reduced, higher numerical precision is achieved, and the method has wide application prospects in the fields of mechanical structure analysis, material mechanical property evaluation, structural dynamics response and the like and is suitable for popularization and application. And particularly, the method has remarkable advantages when a complex structure containing multi-scale characteristics is processed.

Description

Technical Field

[0001] The present invention belongs to the method for simulating the stress field of integrated devices, and specifically relates to a method and system for simulating the stress field of integrated devices based on arbitrary high-order multi-scale local time stepping. Background Art

[0002] As a core technology for the design of integrated devices and the optimization of complex engineering systems, stress field simulation plays a key role in the research of multi-scale physical field coupling. Aiming at the efficiency bottleneck of existing numerical methods, this research proposes an innovative non-uniform spatio-temporal discretization framework: when dealing with multi-scale problems, the traditional time-domain finite difference method not only needs to construct fine grids to represent the microstructure, but is also strictly restricted by the Courant-Friedrichs-Lewy stability criterion, which requires an extremely small time step for iterative calculation in the entire domain, resulting in an exponential increase in the computational cost. The literature Alejandro Cebrecos, Dimitri Krattiger, Victor J. Sánchez-Morcillo, Vicent Romero-García, Mahmoud I. Hussein, The finite-element time-domain method for elastic band-structure calculations, Computer Physics Communications, Volume 238, 2019, Pages 77-87 uses the direct finite element method to establish an elastic mechanics model, and its calculation efficiency is better than that of the traditional time-domain finite difference method, but it still does not solve the problem of low efficiency for multi-scale models. To solve this core technical problem, the present invention proposes a method and system for simulating the stress field of integrated devices based on arbitrary high-order multi-scale local time stepping. Summary of the Invention

[0003] The purpose of the present invention is to provide a method and system for simulating the stress field of integrated devices based on arbitrary high-order multi-scale local time stepping, so as to achieve the optimal allocation of computing resources and higher accuracy.

[0004] The technical solution for realizing the purpose of the present invention is as follows:

[0005] A method for simulating the stress field of integrated devices based on arbitrary high-order multi-scale local time stepping, comprising:

[0006] Step 1, on the non-uniformly partitioned grid of the stress field solution model, expand the stress field using the Taylor series, construct an adaptive time marching algorithm based on the characteristic scale of the grid unit, and iteratively solve the stress field values on the grids in the entire space by establishing a differential time step matching mechanism;

[0007] Step 2: Extract physical parameters related to the stress field according to the calculated stress field values.

[0008] Further, Step 1 specifically includes: First, perform stress field iteration on the grid with the smallest time step. At the intersection of different time steps, use Taylor series expansion to interpolate the stress field values of different time steps required for iteration; after the stress field iteration on the grid with the smallest time step is completed, use Taylor series expansion to calculate the stress field at the interface of the grid with the second smallest time step adjacent to the grid with the smallest time step through interpolation, and then perform stress field iteration on the grid with the second smallest time step; and so on until the stress field iteration on all grids in the entire space is completed.

[0009] Further, the Taylor series expansion is:

[0010]

[0011] where P is the partial differential operator matrix, Δt represents the time step used for iteration, l is the cumulative variable, V(·) is the stress field, and M is the set accuracy order.

[0012] Further, the partial differential operator matrix is:

[0013]

[0014] where the matrices A and B are respectively:

[0015]

[0016]

[0017] where ρ represents the density of the object, E is the elastic modulus, and ν is the Poisson's ratio.

[0018] Further, the difference is:

[0019]

[0020] where n represents the nΔt moment, V c is the stress field of the coarse grid, V f is the stress field of the fine grid, Δt f is the time step of the fine grid, Δt c is the time step of the coarse grid, l, p are cumulative variables, and M is the accuracy order. The stress field value is calculated through this difference formula.

[0021] Further, the grid for non-uniform meshing of the stress field solution model is: Establish an integrated device stress field solution model, use non-uniform grids to discretely mesh the model, and obtain the structural information of the model, including node information and element information on the grids.

[0022] An integrated device stress field simulation system based on arbitrary high-order multi-scale local time stepping, comprising:

[0023] A stress field value solving unit, which expands the stress field by using Taylor series on the non-uniformly meshed grid of the stress field solving model, constructs an adaptive time marching algorithm based on the characteristic scale of the grid unit, and iteratively solves the stress field values on the grids in the entire space by establishing a differential time step matching mechanism;

[0024] A physical parameter determination unit, which extracts the physical parameters related to the stress field according to the calculated stress field values.

[0025] A computer storage medium stores an executable program, and the executable program is executed by a processor to implement the steps of the integrated device stress field simulation method.

[0026] Compared with the prior art, the present invention has the following remarkable advantages: (1) The present invention constructs an adaptive time marching algorithm based on the characteristic scale of the grid unit. By establishing a differential time step matching mechanism, the iteration of the stress field on grids with different time steps can be realized, and the optimal allocation of computing resources can be achieved; (2) The present invention can obtain higher accuracy than the traditional time domain difference method, can enlarge the time step in multi-scale stress field simulation, reduce the time required for calculation, and achieve an obvious acceleration effect compared with the traditional method, providing a new paradigm for large-scale multi-scale complex system simulation. Description of the Drawings

[0027] Figure 1 is the flowchart of the method of the present invention.

[0028] Figure 2 is a schematic diagram of a multi-layer silicon cube model.

[0029] Figure 3 is a schematic diagram of the composition structure of the multi-layer silicon cube model.

[0030] Figure 4 is a stress distribution diagram along the X direction at the observation line (Y = 5 mm, Z = 5 mm). Detailed Embodiments

[0031] The present invention will be further described in detail below with reference to the drawings.

[0032] Combined with Figure 1 , the present invention proposes an integrated device stress field simulation method based on arbitrary high-order multi-scale local time stepping, and the steps are as follows:

[0033] First step: Establish a solution model, discretize the model using non-uniform grids to obtain the structural information of the model, including node information and element information on the grids;

[0034] Second step: On the non-uniformly meshed grids of the stress field solution model, expand the stress field using Taylor series, construct an adaptive time marching algorithm based on the characteristic scale of grid elements, and iteratively solve the stress field values on the grids in the entire space by establishing a differential time step matching mechanism;

[0035] Specifically, this step is as follows: First, perform stress field iteration on the grids with the smallest time step. At the intersection of different time steps, use Taylor series expansion to interpolate the stress field values at different time steps required for iteration. After the stress field iteration on the grids with the smallest time step is completed, use Taylor series expansion to calculate the stress field at the interface of the grids with the second smallest time step adjacent to the grids with the smallest time step through interpolation, and then perform stress field iteration on the grids with the second smallest time step; and so on until the stress field iteration on the grids in the entire space is completed.

[0036] By using the stress field expanded by Taylor series, iteration can be achieved on grids of the same size;

[0037] Starting from the equilibrium differential equation and constitutive equation of elasticity, perform discretization, and then obtain the time-domain recurrence formula.

[0038]

[0039] Among them, v is the velocity component, ρ represents the density of the object, with the unit of (kg / m 3 ) is the Hamiltonian operator, σ is the stress, u is the displacement, ε is the strain, and D is the elastic matrix of the material:

[0040]

[0041] Exchange the left and right terms of equation (1) and deform to obtain:

[0042]

[0043] Expand to:

[0044]

[0045] Let V = [v x , v y , v z , σ xx , σ yy , σ zz , τ xy , τ yz , τxz T , we can obtain the following equation, that is, the iterative differential equation of the stress field:

[0046]

[0047] where P is the partial differential operator matrix, and its form is as follows:

[0048]

[0049] where:

[0050]

[0051]

[0052] Taking the derivative of equation (5) with respect to time, we can obtain By analogy, we can obtain the derivatives of V with respect to time t of all orders. According to the expansion of the Taylor series:

[0053]

[0054] Substituting into equation (5), we get:

[0055]

[0056] where t represents the time instant, and Δt represents the time step used in the iteration. On the same grid, based on equation (5), we can obtain V at any time, and thus the stress field at any time.

[0057] The specific iterative process through interpolation includes:

[0058] Assume that the large time step is Δt = dt, and the small time step is Δt = dt / 5. Iteration is performed with the large time step on the coarse grid and with the small time step on the fine grid.

[0059] (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) we obtain Similarly (here Δt = dt) we obtain Then we obtain successively etc.

[0060] When i0 is at the boundary between the coarse and fine grids, the value of the coarse grid at the previous moment is required when iterating the stress field value at i0. At this time, t = nΔt, which is the initial value of the small time step.

[0061] ​(b), at time t = (n + 0.4)·Δt, update through iteration to obtain When i0 is not at the fine - coarse grid boundary, (where Δt = dt) to obtain Similarly (where Δt = dt) to obtain Then obtain successively etc. When i0 is at the fine - coarse grid boundary, the coarse - grid value of the previous moment is required when iterating the stress - field value at i0. At this time, interpolation is needed:

[0062] After obtaining the coarse - grid value, update the value of i0 at the junction:

[0063]

[0064] Then iterate the stress field at time t = (n + 0.6)·Δt, and so on.

[0065] The third step, data post - processing, extract relevant physical parameters according to the calculated field values.

[0066] This embodiment also provides an integrated device stress - field simulation system based on arbitrary high - order multi - scale local time - stepping, including:

[0067] A stress - field value solving unit, on the grid with non - uniform meshing of the stress - field solving model, expand the stress field using Taylor series, construct an adaptive time - marching algorithm based on the characteristic scale of grid cells, and iteratively solve the stress - field values on the grids in the entire space by establishing a differential time - step matching mechanism;

[0068] A physical - parameter determination unit, extract physical parameters related to the stress field according to the calculated stress - field values.

[0069] This embodiment also provides a computer storage medium, the computer storage medium stores an executable program, and the executable program is executed by a processor to implement the steps of the integrated device stress - field simulation method.

[0070] It should be noted that steps 1 and 3 in the method of the present invention are conventional techniques in the art and will not be described in detail here.

[0071] To verify the correctness and effectiveness of the present invention, the mechanical properties of a multi - layer silicon cube model are analyzed below.

[0072] Use LF - FDTD and ADER - LTS - FDTD algorithms respectively to Figure 2 perform mechanical simulations on the multi - layer silicon cube structure model shown below. The size of this model is 5mm×10mm×10mm, and its composition structure is as Figure 3As shown in the figure. The thickness of the aluminum layer is 3.995 mm, the thickness of the copper layer is 0.005 mm, and the thickness of the silicon layer is 1 mm. The boundary conditions of this model are as Figure 3 shown, and a downward pressure of 1×10 5 N is uniformly applied on the upper surface of the cube, and the other surfaces are set as fixed supports, that is, the displacement is 0. The coarse mesh division size is Δx = 0.05 mm, Δy = 0.4 mm, Δz = 0.4 mm. The fine mesh is distributed in the x direction from 3.995 mm to 4.045 mm. The fine mesh division size in the x direction is Δx = 0.005 mm, and the division size ratio is 10:1. The ratio of the large time step to the small time step is 10:1.

[0073] As Figure 4 shown, the ratio of the total computational amount between the LF-FDTD method and the ADER-LTS-FDTD is 16.7, and the ratio of the time used is 13.2. Considering the additional preprocessing and data exchange operations required during program implementation, the obtained time ratio and computational amount ratio can be considered approximately equal. In addition, the stress distributions obtained by the two methods are approximately equal and are similar to the results of the commercial software COMSOL. The efficiency is increased by 2.213 times compared with COMSOL. Therefore, we can consider that the ADER-LTS method still greatly improves the computational efficiency while ensuring the accuracy when calculating the stress field.

[0074] The method of the present invention performs Taylor series expansion on the stress field in a non-uniform grid, and realizes the accurate description of the stress field distribution through high-order derivative reconstruction technology. Combining the arbitrary order (ADER) technology realizes high-precision time integration, and at the same time introduces the local time stepping (LTS) strategy, enabling different grid regions to adopt different time step sizes, thereby significantly reducing the computational complexity and improving the numerical stability. Compared with the traditional finite difference time domain method, this method not only greatly improves the computational efficiency and reduces the consumption of computational resources, but also realizes higher numerical accuracy, and has broad application prospects in the fields of mechanical structure analysis, material mechanical property evaluation, structural dynamics response, etc., especially showing significant advantages when dealing with complex structures containing multi-scale features.

[0075] 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 stress field simulation method for integrated devices based on arbitrary high-order multi-scale local time stepping, characterized in that: include: Step 1: On the grid of the non-uniformly divided stress field solution model, the Taylor series is used to expand the stress field, and an adaptive time marching algorithm based on the characteristic scale of the grid unit is constructed. By establishing a differentiated time step matching mechanism, the stress field values ​​on the grids in all spaces are iteratively solved; Step 2: Extract physical parameters related to the stress field based on the calculated stress field values.

2. The method for simulating stress field of integrated devices based on arbitrary high-order multi-scale local time stepping according to claim 1, characterized in that: Step 1 specifically includes: first, perform stress field iteration on the grid with the smallest time step, and use Taylor series expansion at the boundary of different time steps to iterate the stress field values ​​of different time steps required by interpolation; after the stress field iteration on the grid with the smallest time step is completed, use Taylor series expansion to interpolate and calculate the stress field at the interface of the grid with the next smallest time step adjacent to the grid with the smallest time step, and then perform stress field iteration on the grid with the next smallest time step; and so on, the stress field iteration on the grids in all spaces is completed.

3. The method for simulating stress field of integrated devices based on arbitrary high-order multi-scale local time stepping according to claim 2, characterized in that: The Taylor series expansion is: Among them, P is the partial differential operator matrix, Δt represents the time step adopted in the iteration, l is the accumulated variable, V(·) is the stress field, and M is the set accuracy order.

4. The method for simulating stress field of integrated devices based on arbitrary high-order multi-scale local time stepping according to claim 1, characterized in that: The partial differential operator matrix is: Among them, matrices A and B are: Among them, ρ represents the density of the object, E is the elastic modulus, and v is the Poisson's ratio.

5. The method for simulating stress field of integrated devices based on arbitrary high-order multi-scale local time stepping according to claim 2, characterized in that: The difference is: Where n represents the time at nΔt, V c is the coarse grid stress field, V f is the fine grid stress field, Δt f is the fine grid time step, Δt c is the coarse grid time step, l and p are cumulative variables, and M is the accuracy order. The stress field value is calculated using the difference formula.

6. The method for simulating stress field of integrated devices based on arbitrary high-order multi-scale local time stepping according to claim 1, characterized in that: The grid of the stress field solution model is non-uniformly divided as follows: a stress field solution model of an integrated device is established, and the model is discretized using a non-uniform grid to obtain the structural information of the model, including node information and unit information on the grid.

7. An integrated device stress field simulation system for implementing the method described in any one of claims 1 to 6, characterized in that: include: The stress field value solving unit uses Taylor series to expand the stress field on the grid of the non-uniform partition of the stress field solving model, constructs an adaptive time-marching algorithm based on the characteristic scale of the grid unit, and iteratively solves the stress field value on the grid in all spaces by establishing a differentiated time step matching mechanism; The physical parameter determination unit extracts physical parameters related to the stress field according to the calculated stress field value.

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 integrated device stress field simulation method according to any one of claims 1 to 6.

Images