High-order multi-scale finite element method and system for chip multi-physics simulation

Abstract

The present invention discloses a high-order multi-scale finite element method and system for chip multi-physics field simulation. The method embeds the designed chip model into a regular coarse background grid, and the embedded simulation will bring about the problem of numerical instability caused by cutting units. To this end, the present invention judges the ill-conditioned coarse nodes of the coarse unit based on the boundary interpolation function; further, the present invention constructs a reduced coarse unit by removing the ill-conditioned boundary interpolation function, thereby ensuring a good condition number. Thereby, a numerical shape function that can reflect the internal material distribution is constructed on the coarse background grid for subsequent simulation. The computational cost is reduced while maintaining the solution accuracy, and the numerical stability of the embedded simulation is improved, which has great potential in the simulation of ultra-large-scale boundary complex structures. It is suitable for multiple simulation fields such as statics, dynamics, and thermal fields, and its numerical stability, simulation accuracy, and computational efficiency advantages have been verified in statics and thermal field simulations.

Description

Technical Field

[0001] The present invention relates to the technical field of embedded simulation of complex boundary structures, and in particular to a numerically stable high-order multi-scale finite element method. Background Art

[0002] Over the past few decades, the finite element method (FEM) has become an important tool in scientific research and engineering, which requires the generation of a boundary-fitting mesh for the considered geometry. However, the generation of boundary-fitting meshes remains a challenging and time-consuming task in many scenarios, including applications that require frequent remeshing (such as fluid-structure interaction and fracture mechanics), or when the geometry is very complex (such as CAD models or CT scan data with clipped surfaces). For these problems, non-fitting finite element methods offer greater advantages because the model is usually embedded in a simple Cartesian background mesh. These methods decouple the geometric domain from the computational domain and avoid the need for complex mesh generation. Non-fitting finite element methods have evolved into various forms, such as the finite element method, embedded finite element, cutFEM method, and cutIGA method.

[0003] In non-conforming finite element methods, standard finite element functions are usually used to approximate the solution, such as Lagrangian shape functions or B-spline / NURBS shape functions. At the same time, the numerical integration strategy needs to be carefully designed to accurately capture the geometric boundaries. The multiscale finite element method (MsFEM) attempts to construct piecewise linear functions, called multiscale shape functions, for modeling heterostructures. By introducing coupling terms, MsFEM is further extended to elliptic problems of vector fields, called EMsFEM. The boundary conditions in the construction of multiscale shape functions have a great influence on the accuracy of the solution. Various boundary conditions have been proposed, such as linear, periodic, and oscillatory boundary conditions. Although EMsFEM shows significant advantages in capturing the interfaces of heterostructures, it is rarely used in non-conforming methods. The main obstacle to the application of multiscale shape functions in non-conforming methods is the existence of cut cells. The multiscale shape functions in these cells that intersect the region are prone to low accuracy and condition number problems.

[0004] In non-conforming finite element methods, the condition number of a linear system depends not only on the element size of the background mesh, but also on the size of the cut elements, which can be arbitrarily small. The condition number problem remains one of the main obstacles in applying non-conforming methods to large-scale problems. However, the condition number problem for multi-scale shape functions (or general piecewise polynomials) in non-conforming methods has not been studied, which is more challenging in large-scale applications and has great potential. This is exactly the focus of this study.

[0005] For a typical large-scale application, such as mechanical and thermal coupled multi-physics simulation of chips, it is a key technology to simulate the performance behavior of chips and related materials under heat transfer, mechanical stress and deformation. The chip model contains small geometric features and material features. Generating a boundary-fitting finite element mesh is a very challenging and time-consuming task, which requires a lot of manual intervention. Moreover, when the design model needs to be frequently modified locally, the mesh needs to be regenerated, which easily introduces low-quality mesh units and discrete errors. Embedded simulation is used to embed the chip model in a regular background grid, and the geometric model is defined by the distance field. It is suitable for dynamic geometry change scenes, does not require regeneration of the mesh, and has a high degree of automation. Embedded simulation seamlessly integrates mechanical, heat conduction and other physical field models. However, embedded simulation inevitably introduces cutting units, even units of arbitrarily small volume, resulting in the problem of arbitrarily large system matrix condition number. The condition number problem has a serious impact on simulation performance in large-scale scenarios. The memory requirements of large-scale scenarios make the direct solution method no longer applicable, and the convergence speed of the iterative solution method is affected by the matrix condition number.

[0006] Based on this, the present invention constructs multi-scale numerical shape functions in a regular background grid, analyzes the system matrix condition number, and proposes a technical solution based on a new interpolation technology to remove ill-conditioned degrees of freedom and construct numerically stable numerical shape functions to improve the stability and accuracy of embedded simulation. Summary of the invention

[0007] In view of the defects of the prior art, the present invention proposes a high-order multi-scale finite element method and system for chip multi-physics field simulation. The present invention analyzes the numerical stability problem of numerical shape functions in cutting units, and proposes a reduction of coarse units on the boundaries of cutting units, ensuring a good condition number, and realizing stable simulation of large-scale chips.

[0008] The technical solution adopted by the present invention is as follows:

[0009] The high-order multiscale finite element method for chip multi-physics simulation includes the following steps:

[0010] 1) The designed chip model is embedded in a regular coarse background grid, and the ill-conditioned coarse nodes of the coarse unit are judged based on the boundary interpolation function;

[0011] 2) Remove the ill-conditioned boundary interpolation function on the coarse element surface, construct a reduced coarse element, ensure a good condition number, and sample the detail points on the coarse grid boundary to obtain the coarse node-boundary temperature transformation matrix;

[0012] 3) By solving the chip temperature field substructure problem within the coarse unit, a temperature mapping relationship from boundary detail points to internal detail points is established to obtain a boundary-internal temperature transformation matrix; further, a coarse node-detail point temperature transformation matrix is ​​obtained, and a numerically stable numerical shape function reflecting the internal material distribution is constructed for each coarse unit for embedded simulation of the chip model.

[0013] In the above technical solution, further, the step 1) judges the ill-conditioned coarse nodes of the coarse unit based on the boundary interpolation function; this is because the condition number of the system matrix of the high-order multi-scale finite element method (hMsFEM) is analyzed and it is found that the boundary interpolation function of the hMsFEM determines its numerical stability; specifically:

[0014] The condition number of the system matrix of the high-order multiscale finite element method hMsFEM is analyzed. for

[0015] ,

[0016] in is the number of coarse units, Coarse unit The unit heat transfer matrix,

[0017] ,

[0018] in is the coarse node-detail point mapping matrix, It is a coarse unit The number of medium and fine grids, It is a fine grid Heat transfer matrix. Heat transfer matrix The condition number is,

[0019] ,

[0020] Because the matrix is a symmetric positive definite matrix, the Euclidean norm above Written as,

[0021] .

[0022] in is the temperature vector corresponding to The function of is a numerical shape function. When there exists a function The norm of Much smaller than the norm hour, becomes very large, resulting in the condition number It will also become very large. This is for a cutting rough unit Upper coarse node Numerical shape function of is likely to happen, for example, its kth component The norm of

[0023] ,

[0024] in It is a thick node The boundary interpolation matrix of The kth column of is the boundary-interior temperature transformation matrix. When the element value of is very small or even 0, for example, there are no boundary detail points in the support domain of the boundary interpolation function, the norm The value of will be very small or even 0. The value of is not affected by the cutting unit. may have arbitrarily large values; on the other hand, This results in the numerical shape function in the rough unit being cut to cause an arbitrarily large condition number. A thick node The condition number of the ill-conditioned node: if the coarse node The corresponding boundary interpolation matrix The maximum value of is less than the given threshold, that is

[0025] ,

[0026] in is a given threshold value, which can be obtained through experiments or experience, usually 0.05, then For coarse elements is a pathologically coarse node, corresponding to is an ill-conditioned numerical shape function.

[0027] Furthermore, the construction of the reduced coarse unit and the boundary-internal temperature transformation in step 2) is as follows:

[0028] Reduced coarse unit Each boundary surface , the coarse nodes are classified into corner points and non-corner points (located on the edge or in the surface), and their subscript sets are recorded as and . Keep all corner points , and well-conditioned non-corner points, whose subscript set is recorded as Then, the surface Any point on The temperature passes through the corner and the retained non-corner points To interpolate,

[0029] ,

[0030] Among them, ij is the coarse node subscript, is the two-dimensional Bernstein basis function, is the interpolation basis function at the corner point. The interpolation function is written in matrix form , traverse all boundary detail points ( is the number of boundary detail points), and the rough node-boundary temperature transformation matrix is ​​obtained ,

[0031] ,

[0032] in is the temperature vector of the coarse node with the subscript ij in the boundary surface F. All the coarse node temperature vectors in the boundary surface F are assembled to form a coarse unit. The temperature vector on .

[0033] Further, step 3) constructs the boundary-internal temperature transformation matrix and the final numerical shape function, as follows:

[0034] By solving the coarse element The chip temperature field substructure problem is to convert the boundary detail point temperature Mapping to internal detail point temperature , which uses the coarse node-boundary temperature transformation as the temperature constraint, and the equilibrium equation is:

[0035]

[0036] in is a submatrix of the heat transfer matrix, and is the subvector of the heat load, which is obtained by solving the second line of the above equation , and assume that ,get:

[0037] ,in ,

[0038] in is the temperature vector of the detail point, is the identity matrix, is the calculated boundary-to-interior temperature transformation matrix.

[0039] Furthermore, the coarse node-edge mapping and boundary-interior mapping Multiply to get the coarse node-detail point temperature transformation matrix ,

[0040] .

[0041] Then, the coarse unit The numerical shape functions on the grid are expressed as piecewise linear shape functions on the fine grid :

[0042] ,

[0043] in It is a linear shape function on a fine grid. The obtained shape function has good numerical stability, satisfies the unit separation, and can reflect the internal material distribution. Its application in embedded simulation can improve numerical stability and obtain high-precision simulation results.

[0044] The core idea of ​​the method of the present invention can be applied to multiple simulation fields such as statics and dynamics in addition to the mechanical and thermal coupling multi-physics field simulation of chips. Therefore, the present invention can also provide a numerically stable high-order multi-scale finite element method for mechanical simulation, including:

[0045] The elastic material model with displacement and load boundary conditions is embedded in a regular coarse background grid, and the ill-conditioned coarse nodes of the coarse element are judged based on the boundary interpolation function.

[0046] Remove the ill-conditioned boundary interpolation function on the coarse element surface, construct a reduced coarse element, and sample the detail points on the coarse grid boundary to obtain the coarse node-boundary displacement transformation matrix;

[0047] By solving the substructure problem in the coarse unit, the displacement mapping relationship between the boundary detail points and the internal detail points is established, and the boundary-internal displacement transformation matrix is ​​obtained;

[0048] Based on the coarse node-boundary displacement transformation and the boundary-internal displacement transformation, a coarse node-detail point displacement transformation matrix is ​​obtained, so as to construct a numerical shape function reflecting the internal material distribution for each coarse element, which is used for embedded simulation of the model.

[0049] The present invention also provides a numerically stable high-order multi-scale finite element simulation system, comprising:

[0050] A node judgment module is used to input the model to be simulated, embed it into the coarse background grid, and judge the ill-conditioned coarse nodes of the coarse units of the model based on the boundary interpolation function;

[0051] The reduced coarse unit construction module constructs the reduced coarse unit according to the ill-conditioned coarse nodes output by the node judgment module, retains all the corner points on the edge and the well-conditioned non-corner points on the surface on the coarse unit surface, and samples the detail points on the coarse grid boundary to obtain the coarse node-boundary temperature transformation matrix;

[0052] The substructure problem solving module solves the substructure problem of the chip temperature field in the coarse unit, establishes the temperature mapping relationship from the boundary detail points to the internal detail points, and obtains the boundary-internal temperature transformation matrix;

[0053] The numerical shape function determination module obtains the coarse node-detail point temperature transformation matrix based on the coarse node-boundary temperature transformation matrix output by the coarse element construction module and the boundary-internal temperature transformation matrix output by the substructure problem solving module, and constructs a numerical shape function that reflects the internal material distribution for each coarse element in a numerically stable manner;

[0054] A simulation module performs embedded simulation on the model based on the obtained numerical shape function.

[0055] The present invention also provides an electronic device, comprising:

[0056] one or more processors;

[0057] A memory for storing one or more programs;

[0058] When the one or more programs are executed by the one or more processors, the one or more processors implement any of the methods described above.

[0059] A computer-readable storage medium stores computer-executable instructions, wherein the instructions are used to implement any of the above methods when executed.

[0060] The beneficial effects of the present invention are:

[0061] 1) The present invention realizes stable simulation of large-scale chip heat transfer problems and can obtain high-precision temperature field results. The matrix condition number of the multi-scale finite element method in embedded simulation is analyzed for the first time, and the conclusion that the boundary interpolation function determines its numerical stability is drawn. Based on this, the ill-conditioned coarse nodes of the coarse units of the large-scale chip model to be simulated are judged, which lays the foundation for stable simulation.

[0062] 2) In order to solve the problem of poor condition number caused by cutting cells, it is proposed to construct reduced coarse cells, which greatly improves the condition number. At the same time, the boundary interpolation function of the reduced coarse cells satisfies the unit divisibility, so that the final constructed numerical shape function also satisfies the unit divisibility, avoiding some non-physical simulation results.

[0063] 3) For each coarse element, the numerical shape function is constructed based on two continuous mappings from coarse nodes to boundary detail points and then to internal detail points. This process only requires solving a small-scale linear system and is computationally efficient. BRIEF DESCRIPTION OF THE DRAWINGS

[0064] Figure 1 In order to reduce the boundary surface of the coarse unit, the positions of the coarse nodes on the three boundary surfaces and the corresponding reduced interpolation functions are shown respectively. The intersection of the three boundary surfaces with the model is different.

[0065] Figure 2 An example of a moving bracket model, (a) the movement of the bracket model, and (b) the coarse grid at t=0 / 0.2 / 0.8.

[0066] Figure 3 The condition number and displacement error versus time curves for the moving bracket model example, including the standard hMsFEM method, the merging strategy, the removing strategy, and the reduction strategy of the present invention.

[0067] Figure 4 Geometry and material definition for large-scale heterogeneous PCB (printed circuit board) models.

[0068] Figure 5 It is the heat conduction simulation result of a large-scale heterogeneous PCB (printed circuit board) model. The picture contains the temperature field distribution and error results of the finite element method and the method of the present invention under different parameters. DETAILED DESCRIPTION

[0069] The present invention is further described below in conjunction with the accompanying drawings and specific examples.

[0070] The core idea of ​​the present invention is to embed the model into a coarse background grid, and by solving the coarse unit boundary substructure problem with displacement boundary conditions, a numerical shape function is constructed for each coarse unit. We solve the ill-conditioned problem of cutting coarse units by constructing a high-order reduction interpolation function for each boundary surface. The method of the present invention can effectively reduce the computational cost and improve the numerical stability of embedded simulation while maintaining the solution accuracy. It has great potential in the simulation of ultra-large-scale boundary complex structures and can be applied to multiple simulation fields such as statics, dynamics, and thermal fields.

[0071] Taking the chip mechanical and thermal coupling multi-physics field simulation as an example, the method of the present invention is specifically as follows:

[0072] The condition number of the system matrix of the high-order multiscale finite element method hMsFEM is mathematically analyzed. for

[0073] ,

[0074] in is the number of coarse units, Coarse unit The unit heat transfer matrix,

[0075]

[0076] in is the coarse node-detail point mapping matrix, It is a coarse unit The number of medium and fine grids, It is a fine grid Heat transfer matrix. Heat transfer matrix The condition number is,

[0077] ,

[0078] in is the Euclidean norm,

[0079] ,

[0080] Because the symmetric matrix The eigenvectors of are orthogonal, and the Euclidean norm is written as,

[0081]

[0082] in and is a matrix The absolute values ​​of the maximum and minimum eigenvalues ​​of . Further, since the matrix is a positive definite matrix, , Moreover, the largest and smallest eigenvalues ​​correspond to the largest and smallest Rayleigh coefficients,

[0083] .

[0084] Based on the function norm and the matrix potential norm The equivalence of can express the maximum and minimum eigenvalues ​​as function norms The Euclidean norm of the corresponding solution vector of business,

[0085] .

[0086] in is the temperature vector corresponding to The function of is a numerical shape function. When there exists a function The norm of Much smaller than the norm hour, becomes very large, resulting in the condition number It will also become very large. This is for a cutting rough unit Upper coarse node Numerical shape function of is likely to happen, for example, its kth component The norm of

[0087] ,

[0088] in It is a thick node The boundary interpolation matrix of The kth column of is the boundary-interior temperature transformation matrix. When the element value of is very small or even 0, for example, there are no boundary detail points in the support domain of the boundary interpolation function, the norm The value of will be very small or even 0. The value of is not affected by the cutting unit. may have arbitrarily large values; on the other hand, This results in the numerical shape function in the rough unit being cut to cause an arbitrarily large condition number. A thick node The condition number of the ill-conditioned node: if the coarse node The corresponding boundary interpolation matrix The maximum value of is less than the given threshold, that is

[0089] ,

[0090] in is a given threshold, which can be 0.05; then For coarse elements is a pathologically coarse node, corresponding to is an ill-conditioned numerical shape function.

[0091] According to the above judgment conditions, the present invention further proposes a reduced coarse unit for improving the condition number problem, as shown below.

[0092] Reduced coarse unit Each boundary surface , the coarse nodes are classified into corner points and non-corner points (located on the edge or in the surface), and their subscript sets are recorded as and . Keep all corner points and well-conditioned non-corner points, and their subscript set is recorded as Then, the surface Any point on The temperature passes through the corner and the retained non-corner points To interpolate,

[0093] ,

[0094] in is the two-dimensional Bernstein basis function, is the interpolation basis function at the corner points, which can be rewritten as

[0095]

[0096] in is a bilinear basis function, is the parameter coordinate of the coarse node. The above reduced interpolation basis function satisfies the unit splitting property.

[0097] ,

[0098] This also makes the final numerical shape function have this property.

[0099] The coarse unit The interpolation function is written in matrix form , traverse all boundary detail points ( is the number of boundary detail points), and the rough node-boundary temperature transformation matrix is ​​obtained ,

[0100] ,

[0101] in It is a coarse unit The coarse nodal temperature vector on .

[0102] Figure 1 The positions of the coarse nodes on the boundary surfaces of the reduced coarse elements and the corresponding reduced interpolation functions are shown, where the intersections of the three boundary surfaces with the model are different.

[0103] On the other hand, the present invention solves the substructure problem in the coarse unit to obtain the boundary-internal temperature transformation matrix, and converts the boundary detail point temperature Mapping to internal detail point temperature , which transforms the coarse node-boundary temperature into temperature constraints and considers the boundary detail points and internal detail points are the retained nodes and the truncated nodes respectively, and the equilibrium equation is:

[0104] ,

[0105] in is a submatrix of the heat transfer matrix, and is the subvector of the heat load, which is obtained by solving the second line of the above equation , and assume that ,get:

[0106] ,

[0107] in is the temperature vector of the detail point, is the identity matrix, is the calculated boundary-to-interior temperature transformation matrix.

[0108] Furthermore, the coarse node-edge mapping and boundary-interior mapping Multiply to get the coarse node-detail point temperature transformation matrix ,

[0109] .

[0110] Then, the coarse unit The numerical shape functions on the grid are expressed as piecewise linear shape functions on the fine grid :

[0111] ,

[0112] in It is a linear shape function on a fine grid. Its application in embedded simulation can improve numerical stability and obtain high-precision simulation results.

[0113] In order to test the performance of the proposed numerical shape function in improving the condition number, it is compared with two other similar numerical stability strategies. The following are two specific solution cases using the method of the core idea of ​​the present invention:

[0114] Test 1 Figure 2 The bracket model shown in (a) is fixed at the bottom and subjected to an external torque, resulting in a tangential traction force on the axis of It moves along the xz plane of the background grid (size 15x8x15). The smallest endpoint (t, 0, t) of the model bounding box is determined by the parameters As the value of t increases, the position of the model relative to the background grid changes, and arbitrarily small cut cells may appear, causing potential conditioning problems. Figure 2 (b) shows the coarse grid at t = 0, 0.2, and 0.8.

[0115] Figure 3The estimated condition number of the system matrix and the simulated displacement error curve are shown at 50 different sampling values ​​of t. Due to the existence of ill-conditioned degrees of freedom, the condition number of the standard MsFEM (Original) is always higher than , which makes the linear system unsolvable. At the same time, it is observed that the condition number of the aggregation strategy is highly sensitive to the position of the model, and the maximum condition number even reaches , which may be due to the fact that many adjacent cutting units form a large aggregate, thereby amplifying the constraint coefficient of the ill-conditioned degrees of freedom and causing the system matrix to be singular. The condition number in the removal strategy is stable and remains very low, below , but its accuracy is very poor, and the maximum displacement error reaches However, the reduction strategy proposed in this invention has significantly improved the condition number and is always lower than , and also shows the highest overall displacement accuracy, with a maximum error of This test example verifies the performance and robustness of the numerical shape function of the present invention in terms of numerical stability.

[0116] Test 2 Figure 4 The simulation is performed on a large-scale heterogeneous PCB (printed circuit board) model shown in Figure 1. The PCB example includes components made of different materials: a fiberglass circuit board, an aluminum heat sink, an alumina ceramic capacitor, a nickel-chromium alloy resistor, a silicon chip and a transistor. A uniform heat source acts on the resistor, chip and transistor components, and the outer surface of the heat sink is affected by convection heat dissipation, with a convection coefficient of , ambient medium temperature ,We test the performance of the proposed method at different coarse grid sizes, 60 x 40 x 12, 75 x 50 x 15, and 120 x 80 x 24, where the global fine grid size is fixed to 600 x 400 x 120. Figure 5 The mesh and simulated temperature fields of the global fine mesh FEM and three test cases are shown. It can be observed that the temperature distributions of the three test cases are close to the benchmark solution, which verifies the ability of the numerical shape function to capture the heat conduction distribution. In addition, as the number of coarse elements increases (i.e., the local mesh size decreases), the temperature error shows a downward trend, with the maximum temperature errors being , ,and . Figure 5 This phenomenon is also shown in the local details in . This can be explained by the following two points: first, the numerical shape function tends to better capture the detailed variation of the material through the smaller size of the local fine mesh; second, the accuracy of the global temperature field tends to improve with the increase of the number of degrees of freedom (DOFs) of the coarse element.

[0117] On the other hand, the total computation time of FEM is 3.7 to 8.0 times that of the present method. In addition, although the linear system solution time increases due to the increase in the number of coarse elements and the increase in the degrees of freedom, the time required for shape function calculation decreases significantly as the local problem size decreases.

[0118] In addition, the present invention also provides a numerically stable high-order multi-scale finite element simulation system, comprising:

[0119] A node judgment module is used to input the model to be simulated, embed it into the coarse background grid, and judge the ill-conditioned coarse nodes of the coarse units of the model based on the boundary interpolation function;

[0120] The reduced coarse unit construction module constructs the reduced coarse unit according to the ill-conditioned coarse nodes output by the node judgment module, retains all the corner points on the edge and the well-conditioned non-corner points on the surface on the coarse unit surface, and samples the detail points on the coarse grid boundary to obtain the coarse node-boundary temperature transformation matrix;

[0121] The substructure problem solving module solves the substructure problem of the chip temperature field in the coarse unit, establishes the temperature mapping relationship from the boundary detail points to the internal detail points, and obtains the boundary-internal temperature transformation matrix;

[0122] The numerical shape function determination module obtains the coarse node-detail point temperature transformation matrix based on the coarse node-boundary temperature transformation matrix output by the coarse element construction module and the boundary-internal temperature transformation matrix output by the substructure problem solving module, and constructs a numerical shape function that reflects the internal material distribution for each coarse element in a numerically stable manner;

[0123] A simulation module performs embedded simulation on the model based on the obtained numerical shape function.

[0124] It will be appreciated by those skilled in the art that embodiments of the present invention may be provided as methods, systems, or computer program products. Therefore, the present invention may take the form of a complete hardware embodiment, a complete software embodiment, or an embodiment combining software and hardware. Furthermore, the present invention may take the form of a computer program product implemented on one or more computer-usable storage media (including but not limited to disk storage, CD-ROM, optical storage, etc.) containing computer-usable program code.

[0125] The present invention is described with reference to flowcharts and / or block diagrams of methods, devices (systems), and computer program products according to embodiments of the present invention. It should be understood that each process and / or block in the flowchart and / or block diagram, as well as the combination of processes and / or blocks in the flowchart and / or block diagram, can be implemented by computer program instructions. These computer program instructions can be provided to a processor of a general-purpose computer, a special-purpose computer, an embedded processor, or other programmable data processing device to produce a machine, so that the instructions executed by the processor of the computer or other programmable data processing device generate instructions for implementing the processes in the flowchart and / or block diagram. Figure 1 A process or multiple processes and / or boxes Figure 1 A device that provides the functions specified in a block or multiple blocks.

[0126] These computer program instructions may also be stored in a computer-readable memory capable of directing a computer or other programmable data processing device to operate in a specific manner, so that the instructions stored in the computer-readable memory produce an article of manufacture comprising an instruction device, which implements the process Figure 1 A process or multiple processes and / or boxes Figure 1 A function specified in one or more boxes.

[0127] These computer program instructions can also be loaded onto a computer or other programmable data processing device so that a series of operating steps are executed on the computer or other programmable device to produce a computer-implemented process, thereby providing instructions for implementing the process. Figure 1 A process or multiple processes and / or boxes Figure 1 The steps for the functions specified in one or more boxes.

[0128] The above are only some of the preferred solutions of the present invention, but they are not intended to limit the present invention. Ordinary technicians in the relevant technical field can make various changes and modifications without departing from the spirit and scope of the present invention, and expand to corresponding electromagnetics, fluid simulation, etc. Therefore, all technical solutions obtained by equivalent replacement or equivalent transformation fall within the protection scope of the present invention.

Claims

1. A high-order multi-scale finite element method for chip multi-physics field simulation, characterized in that: include: The designed chip model is embedded in a regular coarse background grid, and the ill-conditioned coarse nodes of the coarse unit are judged based on the boundary interpolation function; Remove the ill-conditioned boundary interpolation function on the coarse element surface, construct a reduced coarse element, and sample the detail points on the coarse grid boundary to obtain the coarse node-boundary temperature transformation matrix; By solving the chip temperature field substructure problem in the coarse unit, the temperature mapping relationship from the boundary detail point to the internal detail point is established, and the boundary-internal temperature transformation matrix is ​​obtained; A coarse node-detail point temperature transformation matrix is ​​obtained based on the coarse node-boundary temperature transformation matrix and the boundary-internal temperature transformation matrix, and a numerical shape function reflecting the internal material distribution is constructed for each coarse unit to perform embedded simulation on the chip model; Among them: judge the coarse unit A thick node The method to determine whether it is a pathological node is as follows: if the rough node The corresponding boundary interpolation matrix The maximum value of is less than the given threshold, that is ， in For a given threshold, then For coarse elements It is a pathologically thick node; The method for constructing a reduced coarse unit is as follows: Each boundary surface , classify the coarse nodes into corner points on the edge and non-corner points in the surface, and their subscript sets are recorded as and , keep all corner points , and all the well-conditioned non-corner points after removing the ill-conditioned non-corner points, the subscript set is recorded as .

2. The high-order multi-scale finite element method for chip multi-physics field simulation according to claim 1, characterized in that: The boundary surface Any point on The temperature passes through the corner and the retained non-corner points To interpolate, ， Among them, ij is the coarse node subscript, is the two-dimensional Bernstein basis function, is the interpolation basis function at the corner point, and the coarse element The interpolation function is written in matrix form , traverse all boundary detail points , is the number of boundary detail points, and the coarse node-boundary temperature transformation matrix is ​​obtained , ， in is the temperature vector of the coarse node with the subscript ij in the boundary surface F. All the coarse node temperature vectors in the boundary surface F are assembled to form a coarse unit. The temperature vector on .

3. The high-order multi-scale finite element method for chip multi-physics field simulation according to claim 1, characterized in that: By solving the coarse element The chip temperature field substructure problem is to convert the boundary detail point temperature Mapping to internal detail point temperature , which uses the coarse node-boundary temperature transformation as the temperature constraint, and the equilibrium equation is: ， in is a submatrix of the heat transfer matrix, and is the subvector of the heat load, which is obtained by solving the second line of the above equation , and assume that ,get: ,in , in is the temperature vector of the detail point, is the identity matrix, is the calculated boundary-to-interior temperature transformation matrix.

4. The high-order multi-scale finite element method for chip multi-physics field simulation according to claim 1, characterized in that: In the coarse unit In the example, the numerical shape functions are represented as piecewise linear shape functions on a fine grid. : ， in is a linear shape function on the fine grid, It is the coarse node-detail point temperature transformation matrix, that is, the coarse node-boundary mapping and boundary-interior mapping The product of 。 5. A numerically stable high-order multi-scale finite element simulation system, characterized in that: include: A node judgment module is used to input the chip model to be simulated, embed it into the coarse background grid, and judge the pathological coarse nodes of the coarse unit of the model based on the boundary interpolation function; The reduced coarse unit construction module constructs the reduced coarse unit according to the ill-conditioned coarse nodes output by the node judgment module, retains all the corner points on the edge and the well-conditioned non-corner points on the surface on the coarse unit surface, and samples the detail points on the coarse grid boundary to obtain the coarse node-boundary temperature transformation matrix; The substructure problem solving module solves the substructure problem of the chip temperature field in the coarse unit, establishes the temperature mapping relationship from the boundary detail points to the internal detail points, and obtains the boundary-internal temperature transformation matrix; The numerical shape function determination module obtains the coarse node-detail point temperature transformation matrix based on the coarse node-boundary temperature transformation matrix output by the coarse element construction module and the boundary-internal temperature transformation matrix output by the substructure problem solving module, and constructs a numerical shape function that reflects the internal material distribution for each coarse element in a numerically stable manner; A simulation module performs embedded simulation on the model based on the obtained numerical shape function; Among them: judge the coarse unit A thick node The method to determine whether it is a pathological node is as follows: if the rough node The corresponding boundary interpolation matrix The maximum value of is less than the given threshold, that is ， in For a given threshold, then For coarse elements It is a pathologically thick node; The method for constructing a reduced coarse unit is as follows: Each boundary surface , classify the coarse nodes into corner points on the edge and non-corner points in the surface, and their subscript sets are recorded as and , keep all corner points , and all the well-conditioned non-corner points after removing the ill-conditioned non-corner points, the subscript set is recorded as .

6. An electronic device, characterized in that: include: one or more processors; A memory for storing one or more programs; When the one or more programs are executed by the one or more processors, the one or more processors implement the method according to any one of claims 1 to 4.

7. A computer-readable storage medium storing computer-executable instructions, wherein the instructions are used to implement the method according to any one of claims 1 to 4 when executed.

8. A numerically stable high-order multiscale finite element method for mechanical simulation, characterized in that: include: The elastic material model with displacement and load boundary conditions is embedded in a regular coarse background grid, and the ill-conditioned coarse nodes of the coarse element are judged based on the boundary interpolation function. Remove the ill-conditioned boundary interpolation function on the coarse element surface, construct a reduced coarse element, and sample the detail points on the coarse grid boundary to obtain the coarse node-boundary displacement transformation matrix; where: A thick node The method to determine whether it is a pathological node is as follows: if the rough node The corresponding boundary interpolation matrix The maximum value of is less than the given threshold, that is ， in For a given threshold, then For coarse elements It is a pathologically thick node; The method for constructing a reduced coarse unit is as follows: Each boundary surface , classify the coarse nodes into corner points on the edge and non-corner points in the surface, and their subscript sets are recorded as and , keep all corner points , and all the well-conditioned non-corner points after removing the ill-conditioned non-corner points, the subscript set is recorded as ; By solving the substructure problem in the coarse unit, the displacement mapping relationship between the boundary detail points and the internal detail points is established, and the boundary-internal displacement transformation matrix is ​​obtained; A coarse node-detail point displacement transformation matrix is ​​obtained based on the coarse node-boundary displacement transformation matrix and the boundary-internal displacement transformation matrix, so as to construct a numerical shape function reflecting the internal material distribution for each coarse element for embedded simulation of the model.

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