Adaptive mesh refinement techniques for three-dimensional stacked models

1 . A computer-implemented method for adaptive mesh refinement, the method comprising the steps of:

generating a first mesh for a first layer of a stacked model representing a 3D integrated circuit, the first mesh including a first grid for the first layer, the first grid associated with a grid hierarchy, the first grid including a first grid node;

performing a simulation for the stacked model using the first mesh to obtain first results;

calculating an interpolated value for a particular grid node according to the first results, the particular grid node corresponding to a node of the first grid in the grid hierarchy;

generating a second mesh for the first layer of the stacked model, the second mesh including a second grid for the first layer, the second grid associated with the grid hierarchy, and a second grid node of the second grid corresponding to a child grid node of the particular grid node in the grid hierarchy;

performing a simulation for the stacked model using the second mesh to obtain a simulated value of the particular grid node;

generating a comparison between the interpolated value and the simulated value of the particular grid node; and

generating a refined mesh for the first layer including a refined grid associated with the grid hierarchy, wherein the refined grid is generated to include the second grid node based on the comparison.

2 . The computer-implemented method of claim 1 ,

wherein the first grid and the second grid are propagated to each layer of the stacked model, and wherein the interpolated value and the simulated value are calculated for each grid node of the first grid on each layer.

3 . The computer-implemented method of claim 2 , wherein the first mesh, the second mesh, and the refined mesh are generated for each layer of the stacked model.

4 . The computer-implemented method of claim 2 , further comprising determining a maximum error between the interpolated value and the simulated value of the particular grid node across each layer.

5 . The computer-implemented method of claim 4 , wherein the maximum error is determined via a percent error between the interpolated value and the simulated value of the particular grid node across each layer.

6 . The computer-implemented method of claim 4 , wherein comparing the interpolated value and the simulated value for each grid node includes comparing the maximum error to a pre-determined threshold.

7 . The computer-implemented method of claim 6 , wherein the refined mesh omits second grid nodes surrounding the particular grid node if the maximum error is less than the pre-determined threshold.

8 . The computer-implemented method of claim 7 , wherein the refined mesh omits second grid nodes surrounding the particular grid node across all layers if the maximum error is less than the pre-determined threshold.

9 . The computer-implemented method of claim 1 , further comprising:

performing a simulation for the stacked model using the refined mesh to obtain third results;

calculating a further interpolated value for a further particular grid node according to the third results, the further particular grid node corresponding to a node of the second grid in the grid hierarchy;

generating a further mesh for the first layer of the stacked model, the further mesh including a third grid for the first layer, the third grid associated with the grid hierarchy, and a third grid node of the third grid corresponding to a further child grid node of the further particular grid node in the grid hierarchy;

performing a simulation for the stacked model using the further mesh to obtain a further simulated value of the further particular grid node;

generating a further comparison between the further interpolated value and the further simulated value of the further particular grid node; and

generating a further refined mesh for the first layer including a further refined grid associated with the grid hierarchy, wherein the further refined grid is generated to include the third grid node based on the further comparison.

10 . The computer-implemented method of claim 9 , further comprising:

determining that the further refined mesh includes no third grid nodes based on the further comparison; and

outputting the refined mesh as a converged mesh for the stacked model.

11 . A system for adaptive mesh refinement of a stacked model with a plurality of layers, the system including:

a simulation engine for performing simulations on the stacked model using a plurality of meshes with varying refinement levels; and

an adaptive mesh refinement engine for generating the plurality of meshes with an increasing refinement level until a convergence criteria is met, the adaptive mesh refinement engine including:

a grid generation module to propagate successive grids in a grid hierarchy from an initial grid to be used in mesh generation,

a mesh generation module to generate the plurality of meshes including the initial grid and one or more grid points of the successive grids,

an interpolation module to determine an interpolated value for a grid point of the initial grid,

a simulation communication module to provide the plurality of meshes to the simulation engine and to receive a simulated value for the grid point of the initial grid, and

a decision module to compare a difference between the interpolated value and the simulated value to a pre-determined threshold and determine if the convergence criteria is met,

wherein the interpolated value is determined on a first mesh in which each grid point of the initial grid is not included in mesh generation, and wherein the simulated value is determined on a second mesh in which each grid point of the initial grid is included in mesh generation.

12 . The system of claim 11 , wherein the adaptive mesh refinement engine further includes a layering module to propagate the successive grids and plurality of meshes throughout each layer of the stacked model.

13 . The system of claim 12 , wherein the decision module includes a maximum error calculator to determine a maximum error between the interpolated value and the simulated value for the grid point across each layer of the stacked model.

14 . The system of claim 13 , wherein the decision module compares the maximum error to the pre-determined threshold to determine if the convergence criteria is met for the grid point across all layers of the stacked model.

15 . The system of claim 11 , wherein the adaptive mesh refinement engine further includes a rollback module to remove one or more refined grid points from the second mesh if the convergence criteria is met in the decision module for the grid point.

16 . The system of claim 11 , wherein the initial grid is selected from the group consisting of a user-defined grid, a power-based hard point grid, a hotspot-based hard point grid, a hard point grid uniformly-distributed about one or more components of the stacked model, a machine-learning generated hard point grid, and any combination thereof.

17 . The system of claim 11 , wherein the adaptive mesh refinement engine further comprises an output module to output a converged mesh when the convergence criteria is met, and wherein the convergence criteria is met when the difference between the interpolated value and the simulated value is less than the pre-determined threshold at each grid point.

18 . A computer-implemented method of performing mesh refinement on a stacked model including an initial grid on a plurality of layers to define layered grid points, the method comprising:

receiving a first dataset from a first simulation in which the initial grid is not included in a simulated mesh;

calculating an interpolated value for each grid point of the initial grid using the first dataset;

receiving a second dataset from a second simulation in which the initial grid is included in the simulated mesh and a further grid is generated including child nodes branching from each layered grid point of the initial grid;

calculating a maximum error between the interpolated value and a simulated value of the second dataset for each layered grid point of the initial grid across the plurality of layers;

comparing the maximum error of each layered grid point to a pre-determined threshold of mesh convergence;

removing converged child nodes of the further grid corresponding to each layered grid point of the initial grid that has a maximum error less than the pre-determined threshold to generate a cleaned grid; and

generating a refined mesh from the cleaned grid, wherein the refined mesh includes each grid point of the cleaned grid, and wherein a refined grid is generated including child nodes branching from each grid point of the cleaned grid.

19 . The computer-implemented method of claim 18 , further comprising:

receiving a third dataset from a third simulation using the refined mesh as the simulated mesh;

calculating a further interpolated value for each grid point of the cleaned grid using the second dataset;

calculating a maximum error of each layered grid point of the cleaned grid between the further interpolated value and a simulated value of the third dataset; and

comparing the maximum error of each layered grid point of the cleaned grid to the pre-determined threshold of mesh convergence.

20 . The computer-implemented method of claim 19 , further comprising:

removing further converged child nodes of the refined grid corresponding to each layered grid point of the cleaned grid that has a maximum difference less than the pre-determined threshold to generate a further cleaned grid.