Electromagnetic field solving method based on finite element and boundary element in EDA

Abstract

The invention discloses an electromagnetic field solving method based on finite elements and boundary elements in EDA. According to the method, by combining the advantages of a finite element method (FEM) and a boundary element method (BEM), rapid and accurate solving of electromagnetic field distribution in an integrated circuit is achieved. The method comprises the following steps: step 1, constructing a geometric model of an integrated circuit; 2, carrying out regional discretization and primary function approximation; 3, solving an algebraic equation by a numerical method; step 4, integrating internal and boundary electromagnetic field solving results through a coupling algorithm; and 5, precision correction and result visualization are carried out. According to the method, the electromagnetic field solving precision and efficiency can be remarkably improved, the design period is shortened, and the method is suitable for various complex circuit designs.

Description

Technical Field

[0001] The present invention relates to the technical field of electronic design automation (EDA), and in particular to an electromagnetic field solving method based on finite element and boundary element coupling in EDA. Background Art

[0002] With the development of integrated circuit scale and the progress of electronic design automation, the characteristic size of integrated circuit (IC) is getting smaller and smaller, the operating frequency is gradually increasing, and the electromagnetic effect is becoming more and more significant in circuit design. Under high frequency and high speed conditions, the influence of parasitic resistance, capacitance, inductance and electromagnetic radiation effects on circuit performance is particularly obvious, and parasitic parameter extraction has become an important part of circuit design performance analysis and improvement.

[0003] The purpose of parasitic parameter extraction is to create an accurate simulation model of the circuit for subsequent timing analysis, power analysis, circuit simulation, and signal integrity analysis, and to help designers perform iterative optimization and process improvement. High-precision parasitic parameter extraction relies on solving Maxwell's equations. Therefore, solving and evaluating the electromagnetic field of large-scale integrated circuits is a prerequisite for component parameter extraction.

[0004] Traditional electromagnetic field solution methods are usually based on detailed physical information extracted from IC layout files, such as the size and layout of components such as transistors, resistors, and capacitors, and use numerical methods to build circuit models for solution. Finite element method (FEM) and boundary element method (BEM) are two common traditional solution methods. Among them, the finite element method proposes a relatively complete numerical solution approximation method for complex geometric structures, while the boundary element method further improves the finite element method in complex boundary fields and reduces the dimension of calculation.

[0005] However, in large-scale integrated circuits, the amount of calculation of the finite element method increases dramatically. Although the boundary element method reduces the calculation area, the processing of complex three-dimensional structures and irregular boundaries still requires a lot of computing resources. At the same time, there are multi-scale structures from nanoscale to macroscale in integrated circuits, and traditional single methods are difficult to process these complex structures of different scales at the same time. Summary of the invention

[0006] Based on the above background, the present invention couples the two classical methods of finite element method and boundary element method to help solve complex electromagnetic field problems in EDA accurately and efficiently. The present invention improves the calculation efficiency and overcomes the problems of low flexibility, large amount of calculation, and inaccurate numerical values ​​brought by a single numerical method by dividing different regions. The present invention has greater flexibility in dealing with complex circuit designs and different actual indicator requirements, and has higher solution accuracy in simulations of high-frequency environments.

[0007] The technical solution of the present invention is specifically described as follows.

[0008] The present invention provides an electromagnetic field solving method based on finite element and boundary element in EDA, comprising the following steps:

[0009] Step 1: construct a 2D geometric model of the integrated circuit. The 2D geometric model is used to simulate different integrated circuit structures and corresponding electromagnetic field distribution types, which includes different dielectric layers, metal interconnects and contact hole information in the integrated circuit;

[0010] Step 2: Finite element and boundary element region division

[0011] The extracted geometric model is preliminarily divided according to the geometric structure characteristics and the distribution characteristics of the electromagnetic field to obtain the finite element region for finite element method processing and the boundary element region for boundary element method processing;

[0012] Step 3: Establishment of discrete grid

[0013] Different discrete grid densities are used in the finite element area and the boundary element area to achieve a balance between accuracy and efficiency;

[0014] Step 4: Basis function simulation

[0015] Determine the type of basis function in each discrete unit, perform linear combination of the basis functions of all discrete units, apply Maxwell's laws, and construct a global algebraic equation system; based on the finite element and boundary element regions divided in step 2, apply the finite element and boundary element numerical methods to solve the algebraic equations and obtain a preliminary numerical approximation of the electromagnetic field;

[0016] Step 5: Coupling

[0017] The accuracy and stability of the calculation results of the finite element and boundary element regions are checked through the coupling algorithm, and then the regional grid and basis function are optimized according to the situation, and the secondary solution of the electromagnetic field distribution is achieved through iteration; among which:

[0018] The specific coupling algorithm is as follows: check the continuity of the value and rate of change of the electromagnetic field at the boundary between the finite element region and the boundary element region. The numerical solutions of the electromagnetic fields in the finite element region and the boundary element region are denoted as E FEM and E BEM , calculate the residual on the boundary:

[0019] ΔE=E FEM -E BEM

[0020] Set a threshold ∈. When the boundary residual is greater than the threshold, that is, ||ΔE||>∈, iterative optimization is required.

[0021] In the present invention, in step 1, the method for constructing a geometric model of an integrated circuit is as follows:

[0022] (1) Clarify the performance indicators, functional requirements and physical limitations of integrated circuits;

[0023] (2) EDA tools create the preliminary design of the circuit, including defining the size and layout of components such as transistors, resistors, and capacitors;

[0024] (3) Based on the preliminary design, detailed layout design is carried out, including the precise location of components, the routing of interconnection lines, and the planning of power and ground networks;

[0025] (4) Perform design rule checks on the above designs to ensure that the minimum feature size, wiring width, and spacing requirements under the given manufacturing process are met;

[0026] (5) Use simulation tools to make a preliminary estimate of circuit performance and iteratively optimize the design as appropriate to form a complete 2D geometric model that meets the requirements.

[0027] In the present invention, in step 2, the finite element region processed by the finite element method includes regions with complex geometric shapes and internal details, such as three-dimensional interconnected structures, irregularly shaped conductors and dielectric regions, and regions that require detailed analysis, such as regions with concentrated current or large changes in electric field strength, which are suitable for capacitors, inductors or transistor structures;

[0028] The boundary element area processed by the boundary element method includes the external boundary of the integrated circuit, as well as problems with large gradient changes in boundary variables, such as stress concentration or crack problems. It is suitable for the electric field distribution of chip external packaging and interconnect lines.

[0029] In the present invention, in step three, the grid division rule is generally processed according to the rule that the high gradient area is dense and the low gradient area is sparse. In the design of integrated circuits with micron-level feature sizes, it is recommended to set the grid side length to 1 / 10 to 1 / 5 of the feature size in the finite element FEM area and 1 / 5 to 1 / 2 in the boundary element BEM area, and ensure that the ratio of the discrete grid unit size at the BEM boundary to the unit size near the boundary in the FEM area is close to 1 in the coupling area under ideal conditions, and control the ratio of the BEM to FEM unit size to not exceed 2:1, so as to avoid discontinuity and instability of the numerical solution at the boundary during coupling.

[0030] In the present invention, in step 5, a coupling algorithm is used to accurately couple the calculation results of the finite element and the boundary element, and a coupling matrix is ​​constructed by mixing boundary conditions, and the stability and convergence of the calculated values ​​and the compliance with boundary conditions, such as the continuity of the electromagnetic field, are checked in turn. When the convergence conditions are not met, the division of the discrete grid and the selection of the basis function are further optimized for the coupling region, wherein the optimization of the grid unit follows the grid size specification in step 3 to ensure the convergence speed of the secondary solution.

[0031] Compared with the prior art, the present invention has the following advantages:

[0032] (1) Finite element and boundary element are organically combined, and the internal and external electromagnetic fields can be solved by relatively independent sets of equations, which reduces the computational scale and the number of redundant iterations. Especially in high-frequency problems, the discretization of boundary elements can effectively reduce the complexity of boundary conditions, while the efficient solution of finite elements in the internal area ensures the overall accuracy.

[0033] (2) Optimization of computing resource consumption. The computing resources required for the discretization of boundary elements are much less than those for the global discretization of finite elements, making it more suitable for complex integrated circuit (IC) design.

[0034] (3) The finite element and boundary element coupling method has good adaptability to structures of different scales and can achieve multi-scale coverage from nanoscale to macroscale while maintaining a balance between computational efficiency and accuracy; the high-precision electromagnetic field numerical solution of the present invention provides an accurate numerical basis for the parasitic parameter extraction of the field solver. BRIEF DESCRIPTION OF THE DRAWINGS

[0035] Figure 1 : A flowchart of an electromagnetic field solution method based on finite elements and boundary elements in EDA of the present invention.

[0036] Figure 2 : Some examples of integrated circuit geometric models.

[0037] Figure 3 :Region division of finite element and boundary element.

[0038] Figure 4 : Schematic diagram of the geometric model generation process.

[0039] Figure 5 :Schematic diagram of the process of establishing algebraic equations. DETAILED DESCRIPTION

[0040] The present invention provides an electromagnetic field solution method based on finite element and boundary element in EDA, the principle of which is as follows:

[0041] As a spatial continuous field, the electromagnetic field is characterized by partial differential equations. Using the finite element and boundary element methods, the partial differential equations can be converted into approximate linear equations with the help of discretized grids, and the electromagnetic field at any point can be approximated by the linear combination of grid basis functions. This method can be used to solve various unknown parameters in the electromagnetic field problem, including electric field intensity, potential, magnetic field intensity, magnetic potential, charge density, current density, etc. h For example, the actual electric field can be approximately expressed as follows:

[0042]

[0043] where each i represents a cell in the discrete grid, ψ i is the basis function selected in the discretization step, E i Represents E used to approximate the actual electric field h The coefficient of the linear term in the function.

[0044] The above principle of constructing linear equations is applicable to both finite element and boundary element methods. Since boundary element only deals with regional boundary problems, the dimension of the problem can be reduced by one order to improve computational efficiency. Specifically, this design is a one-dimensional problem, thereby enhancing the ability to handle complex boundaries.

[0045] like Figure 1 As shown, the present invention comprises the following steps:

[0046] Construct 2D geometric models of integrated circuits to simulate different integrated circuit structures and corresponding electromagnetic field distribution types; Figure 2 The figure shows some examples of integrated circuit geometry models. The model includes information such as different dielectric layers, metal interconnects, contact holes, etc. in the integrated circuit. This information is helpful for the subsequent extraction of electromagnetic fields.

[0047] Regional discretization and basis function approximation are used to approximate the electromagnetic field distribution in different regions, so as to facilitate the application of finite element and boundary element methods. Preprocessing the geometric model of the circuit is a key step of the present invention, identifying the regions suitable for finite element method and boundary element method respectively, and finding appropriate discrete grids and basis functions to simulate the electromagnetic field in the discretized unit. Figure 3 As shown in the figure, the components inside a circuit after extraction can be preliminarily divided into dense mesh finite element area (FEM1 area), ordinary mesh finite element area (FEM2 area), and the boundary between finite element and boundary element (FEM-BEM Boundary) according to their geometric model, geometric structure complexity and electromagnetic field distribution characteristics. The division of different areas helps to improve the calculation efficiency and overcome the problems of low flexibility, large calculation amount, and numerical inaccuracy brought by a single numerical method.

[0048] Establish a discrete grid, transform the continuous mathematical model of the electromagnetic field into a discrete numerical model, and apply numerical methods to solve algebraic equations;

[0049] The internal and boundary electromagnetic field solution results are integrated through the coupling algorithm; the coupling algorithm is an algorithm for accuracy checking and optimization of the numerical solutions of the electromagnetic fields in the two regions initially obtained. Its main purpose is to determine whether a secondary solution is needed to reduce the error, and to return the optimized discrete grid and basis function if iteration is required.

[0050] The results are corrected for accuracy and visualized. If necessary, the distribution and density of the electromagnetic field basis functions and discrete grids are adjusted to obtain accurate and visualized electromagnetic field distribution.

[0051] The following is a detailed description of the implementation examples of the present invention:

[0052] Example 1

[0053] In Example 1, we first give a specific design indicator to be achieved and generate a preliminary geometric model, which is the basis for subsequent finite element and boundary element electromagnetic field solution methods. Figure 4 , the specific steps are:

[0054] Provide design indicators: Designers first need to provide the performance indicators, functional requirements, implementation process and physical limitations of the generated integrated circuit;

[0055] Automated preliminary design: EDA tools create a preliminary design of the circuit, including defining the size and layout of components such as transistors, resistors, and capacitors;

[0056] Refined layout design: Based on the preliminary design, detailed layout design is carried out, including the precise location of components, the routing of interconnection lines, and the planning of power and ground networks;

[0057] Design rule check: Perform design rule check on the above design to ensure that the minimum feature size, wiring width and spacing requirements under the given manufacturing process are met;

[0058] Performance evaluation and optimization: Use simulation tools to make a preliminary estimate of circuit performance and appropriately iteratively optimize the design to form a complete 2D geometric model that meets the requirements.

[0059] Example 2

[0060] In Example 2, the generated geometric model is used to first divide the finite element and boundary element regions, and discretize the respective regions to facilitate the application of finite element and boundary element solutions and coupling algorithms; the flow chart is as follows: Figure 5 ;

[0061] Finite element and boundary element region division: For the extracted geometric model, the region can be divided according to the complexity of the geometric structure and the distribution characteristics of the electromagnetic field, and processed by the finite element method and boundary element method respectively. The extracted geometric model is imported into the layout software for circuit component extraction and analysis, and the divided finite element and boundary element regions can be preliminarily generated according to different components. In the subsequent iterative optimization process, the coupling algorithm will further optimize the divided region according to the convergence and stability of the calculation results, as well as whether the boundary conditions are met.

[0062] Establishment of discrete grids: Depending on whether the electromagnetic field changes are dense and whether detailed analysis is required, different discrete grid densities should be adopted in the finite element area and the boundary element area to achieve a balance between accuracy and efficiency. Designers can pre-set the discretized grid density, the number of discrete units, and the shape of the grid units. These parameters can be manually adjusted according to the accuracy and operation speed. In the automated processing, the circuit model is covered by unstructured grids by default, such as triangular or tetrahedral grids. Areas with lower precision requirements can be covered only roughly according to manual needs to save calculations.

[0063] Basis function simulation: Determine the basis function type in each discrete unit. The basis function type is given by the common basis function library, mainly including linear approximation and quadratic approximation. Linearly combine the basis functions of all discrete units and apply Maxwell's laws to construct a global algebraic equation system for solving the numerical solution of the electromagnetic field and checking and optimizing the coupling algorithm. The coupling algorithm iteratively optimizes the selection of basis functions by estimating the error and evaluating the boundary conditions.

[0064] After the above steps, we have preliminarily obtained a global set of equations, and solving them numerically can give us a preliminary numerical approximation of the electromagnetic field.

[0065] The coupling algorithm can be used to check the accuracy and stability of the calculation results of the finite element and boundary element regions, and provide certain iterative optimization information for secondary solution according to the situation. The specific steps of the algorithm are: first input the existing geometric model, regional meshing and set basis functions, and input the numerical solution of the electromagnetic field in the finite element and boundary element regions. The algorithm first checks the continuity of the value and rate of change of the electromagnetic field at the boundary between the finite element and the boundary element. The numerical solutions of the electromagnetic field in the finite element and boundary element regions are E FEM and E BEM , calculate the residual on the boundary:

[0066] ΔE=E FEM -E BEM

[0067] Set a threshold ∈. When the boundary residual is greater than the threshold, that is, ||ΔE||>∈, it means that iterative optimization is required.

[0068] The iterative optimization strategy is based on the residual distribution calculated in the previous step, and is divided into regional mesh optimization and basis function optimization. For regional meshes, mesh encryption is performed in areas with large residuals; at the same time, local mesh refinement is performed for cells with high errors inside the finite element region. After local mesh encryption, the entire finite element or boundary element region is re-meshed and a new network model is returned. For basis functions, the order of basis functions can be appropriately increased in local areas with large errors, different basis function types can be adaptively selected, or mixed basis functions and basis function orthogonal optimization can be performed in local complex fields. Multiple basis functions are used to improve accuracy, while low-order basis functions are retained in other areas to balance the amount of calculation.

[0069] Example 3

[0070] In Example 3, we integrated the coupling algorithm into the example test, set the 3D stacked IC design indicators (including multi-layer metal interconnection, packaging and external connection) and different design processes, and performed electromagnetic field simulation analysis on the generated design using finite element and boundary element coupling, and compared the performance of our coupling method with the traditional finite element method in electromagnetic field solution.

[0071] Design indicators include: 8 layers of metal interconnection, in which the upper layer of metal is the power supply, ground and clock network, the lower layer of metal is used for signal transmission, the stacked chips are connected by micro bumps and through silicon vias (TSV), and the open boundary area is used to evaluate electromagnetic radiation and far-field effects.

[0072] The simulation results at different frequencies show that: for the electromagnetic field distribution at low frequency (1GHz), both methods can achieve similar calculation accuracy (relative error <1%), and both can accurately capture the local electromagnetic field distribution; for the electromagnetic field distribution at high frequency (20GHz), the traditional finite element method has numerical oscillations in the boundary area, resulting in distortion, while the coupling method is more stable in the boundary area, and the boundary element has more advantages in dealing with far-field radiation and scattering. The simulation results are closer to the actual situation, with an error of less than 2%. In terms of computational complexity, the coupling algorithm under the default parameters can reduce the number of discrete grids to less than 50% of the traditional finite element method, and the memory usage is reduced to 70% of the traditional method. At the same time, due to the introduction of the coupling algorithm, the coupling method has better convergence and fewer iterations, which brings a 14-fold speed increase in solution efficiency.

[0073] The test results show that the solution method of finite element and boundary element coupling has greater flexibility in dealing with complex circuit design and different actual index requirements, and has higher solution accuracy in the simulation of high-frequency environment. At the same time, this design can be used as an effective step for subsequent field solvers to extract parasitic parameters, and overall improve the efficiency of circuit design and simulation of complex 3D structures and multi-scale problems.

Claims

1. An electromagnetic field solution method based on finite element and boundary element in EDA, characterized in that: The following steps are involved: Step 1: construct a 2D geometric model of the integrated circuit. The 2D geometric model is used to simulate different integrated circuit structures and corresponding electromagnetic field distribution types, which includes different dielectric layers, metal interconnects and contact hole information in the integrated circuit; Step 2: Finite element and boundary element region division The extracted geometric model is preliminarily divided according to the geometric structure characteristics and the distribution characteristics of the electromagnetic field to obtain the finite element region for finite element method processing and the boundary element region for boundary element method processing; Step 3: Establishment of discrete grid Different discrete grid densities are used in the finite element area and the boundary element area to achieve a balance between accuracy and efficiency; Step 4: Basis function simulation Determine the type of basis function in each discrete unit, perform linear combination of the basis functions of all discrete units, apply Maxwell's laws, and construct a global algebraic equation system; based on the finite element and boundary element regions divided in step 2, apply the finite element and boundary element numerical methods to solve the algebraic equations and obtain a preliminary numerical approximation of the electromagnetic field; Step 5: Coupling The accuracy and stability of the calculation results of the finite element and boundary element regions are checked through the coupling algorithm, and then the regional grid and basis function are optimized according to the situation, and the secondary solution of the electromagnetic field distribution is achieved through iteration; among which: The specific coupling algorithm is as follows: check the continuity of the value and rate of change of the electromagnetic field at the boundary between the finite element region and the boundary element region. The numerical solutions of the electromagnetic fields in the finite element region and the boundary element region are denoted as E FEM and E BEM , calculate the residual on the boundary: ΔE=E FEM -E BEM Set a threshold ∈. When the boundary residual is greater than the threshold, that is, ||ΔE||>∈, iterative optimization is required.

2. The electromagnetic field solution method based on finite element and boundary element in EDA according to claim 1, characterized in that: In step 1, the method of constructing a 2D geometric model of an integrated circuit is as follows: (1) Clarify the performance indicators, functional requirements and physical limitations of integrated circuits; (2) EDA tools create a preliminary design of the circuit, including defining the size and layout of transistors, resistors, and capacitors; (3) Based on the preliminary design, detailed layout design is carried out, including the precise location of components, the routing of interconnection lines, and the planning of power and ground network; (4) Perform design rule checks to ensure that the minimum feature size, wiring width, and spacing requirements are met under a given manufacturing process; (5) Use simulation tools to make a preliminary estimate of circuit performance, and then iteratively optimize the design based on the results to form a 2D geometric model that meets the requirements.

3. The electromagnetic field solution method based on finite element and boundary element in EDA according to claim 1, characterized in that: In step 2, the finite element area processed by the finite element method includes three-dimensional interconnected structures, irregularly shaped conductor and dielectric areas, and capacitor, inductor or transistor structure areas; the boundary element area processed by the boundary element method includes the external boundary of the integrated circuit, and the electric field distribution area of ​​the chip external packaging and interconnection lines.

4. The electromagnetic field solution method based on finite element and boundary element in EDA according to claim 1, characterized in that: In step three, a discrete grid is established based on the rule that high gradient areas are dense and low gradient areas are sparse.

5. The electromagnetic field solution method based on finite element and boundary element in EDA according to claim 1, characterized in that: In step 4, the basis function type is given by the basis function library, including linear approximation and quadratic approximation.

6. The electromagnetic field solution method based on finite element and boundary element in EDA according to claim 1, characterized in that: In step 4, in the coupling region, the ratio of the discrete grid unit size at the BEM boundary to the unit size near the boundary in the FEM region is ideally close to 1, and the ratio of the BEM to FEM unit size is controlled to be no more than 2:1 to avoid discontinuity and instability of the numerical solution at the boundary during coupling.

7. The electromagnetic field solution method based on finite element and boundary element in EDA according to claim 1, characterized in that: In step five, the optimization of the regional mesh includes: mesh encryption in the area with large residuals; local mesh refinement of cells with high errors inside the finite element area; re-meshing the entire finite element or boundary element area after local mesh encryption, and returning a new network model.

8. The electromagnetic field solution method based on finite element and boundary element in EDA according to claim 1, characterized in that: In step five, the optimization of basis functions includes: increasing the order of basis functions in local areas with large errors, adaptively selecting different basis function types, or performing mixed basis function and basis function orthogonal optimization in local complex fields, using multiple basis functions to improve accuracy, and retaining low-order basis functions in other remaining areas to balance the amount of calculation.

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