Numerical simulation method for analyzing transient thermal stress of electronic packaging system

Abstract

The invention provides a numerical simulation method for analyzing transient thermal stress of an electronic packaging system, and the method comprises the steps: building a temperature field equation and boundary conditions of a Laguerre domain based on a Laguerre polynomial basis function; establishing a stress field equation and boundary conditions of a Laguerre domain based on the Laguerre polynomial basis function; setting material attributes, heat sources and boundary conditions of the packaging system, and respectively establishing Laguerre coefficient matrix equations of temperature and displacement based on a finite element method; the matrix equation is solved step by step, and Laguerre coefficients of all orders of temperature and displacement are obtained in sequence; based on a Laguerre coefficient, reconstructing time domain temperature and displacement distribution of the packaging system; and calculating the transient thermal stress of the packaging system according to the time domain temperature and the displacement result. The method has the advantages of unconditional stability and high efficiency on the premise of ensuring the precision.

Description

Technical Field

[0001] The present invention relates to the field of numerical simulation computing technology, and in particular to a numerical simulation method for analyzing transient thermal stress of an electronic packaging system. Background Art

[0002] The continuous increase in power consumption of electronic devices poses a severe challenge to the thermal management of electronic packaging systems. At the same time, due to high temperatures and the mismatch of mechanical parameters between different materials, the stability of the packaging system is greatly reduced and the service life is shortened. Therefore, it is crucial for engineers in the early design stage to accurately and efficiently obtain the temperature and stress distribution of the packaging system.

[0003] Compared with the more expensive experimental methods, numerical simulation methods can conveniently and intuitively provide detailed information on the temperature and stress of electronic packaging systems. However, for transient thermal stress simulation problems, the stability and accuracy of numerical simulation methods are closely related to the time step set in the simulation. When the time step is small, although the stability of the algorithm can be guaranteed, it will cause the simulation time to be too long; on the other hand, when the time step is set to a large size, the simulation time can be effectively shortened, but it usually faces a series of serious problems such as deterioration of calculation accuracy and loss of algorithm stability. Therefore, for the transient thermal stress analysis scenario of the packaging system, it is necessary to develop an efficient numerical simulation method to solve the above bottleneck problems. Summary of the invention

[0004] In view of the defects in the prior art, the purpose of the present invention is to provide a numerical simulation method for analyzing transient thermal stress of an electronic packaging system. The method of the present invention has the advantages of unconditional stability and high efficiency while ensuring accuracy.

[0005] To solve the above problems, the technical solution of the present invention is:

[0006] A numerical simulation method for analyzing transient thermal stress of an electronic packaging system comprises the following steps:

[0007] Based on Laguerre polynomial basis functions, the temperature field equation and boundary conditions of the Laguerre domain are established;

[0008] Based on Laguerre polynomial basis functions, the stress field equations and boundary conditions of the Laguerre domain are established;

[0009] Set the material properties, heat sources and boundary conditions of the packaging system, and establish the Laguerre coefficient matrix equations of temperature and displacement based on the finite element method;

[0010] Solve the matrix equations step by step to obtain the Laguerre coefficients of temperature and displacement in turn;

[0011] Based on the Laguerre coefficient, the time domain temperature and displacement distribution of the packaging system are reconstructed;

[0012] Based on the time-domain temperature and displacement results, the transient thermal stress of the packaging system is calculated.

[0013] Preferably, the step of establishing the temperature field equation and boundary conditions of the Laguerre domain based on the Laguerre polynomial basis function specifically includes:

[0014] In the first step, the temperature variable is expanded in the time domain using the Laguerre basis function;

[0015] The second step is to substitute the expression obtained in the first step into the time-domain temperature field equation, and use the orthogonality of the Laguerre basis function to eliminate the time term, and obtain the temperature field governing equation with the temperature Laguerre coefficient as the unknown quantity;

[0016] In the third step, the expression obtained in the first step is substituted into the temperature field boundary condition, and the time term is eliminated by using the orthogonality of the Laguerre basis function to obtain the temperature field boundary condition with the temperature Laguerre coefficient as the unknown quantity.

[0017] Preferably, the step of establishing the stress field equation and boundary conditions of the Laguerre domain based on the Laguerre polynomial basis function specifically includes:

[0018] In the first step, the displacement variables are expanded in the time domain using Laguerre basis functions;

[0019] In the second step, the expression obtained in the first step is substituted into the stress field equation in the time domain, and the time term is eliminated by using the orthogonality of the Laguerre basis function to obtain the stress field governing equation with the displacement Laguerre coefficient as the unknown quantity;

[0020] In the third step, the expression obtained in the first step is substituted into the stress field boundary condition, and the time term is eliminated by using the orthogonality of the Laguerre basis function to obtain the stress field boundary condition with the displacement Laguerre coefficient as the unknown quantity.

[0021] Preferably, the step of setting the material properties, heat sources and boundary conditions of the packaging system and establishing the Laguerre coefficient matrix equation of temperature and displacement based on the finite element method specifically includes:

[0022] The first step is to set the material properties, heat source, temperature field boundary conditions and stress field boundary conditions of the packaging system;

[0023] In the second step, combined with the parameter setting in the first step, the finite element method is used to discretize the space and the matrix equation of the temperature Laguerre coefficient is obtained;

[0024] In the third step, combined with the parameter settings in the first step, the finite element method is used to discretize the space and the matrix equation of the displacement Laguerre coefficient is obtained.

[0025] Preferably, in the step of solving the matrix equations step by step to obtain the Laguerre coefficients of each order of temperature and displacement in turn, the step-by-step solution refers to gradually increasing the order of the Laguerre basis function from the 0th order to obtain the corresponding matrix equation, and using a large sparse matrix solver to solve the equation group to obtain the Laguerre coefficients of each order.

[0026] Preferably, the step of reconstructing the time-domain temperature and displacement distribution of the packaging system based on the Laguerre coefficient specifically includes:

[0027] In the first step, based on the obtained temperature and displacement Laguerre coefficients and combined with the Laguerre basis function expansion formula, the time domain temperature and displacement distribution of the packaging system are reconstructed;

[0028] In the second step, data filtering is used to reduce the Laguerre order required to reconstruct the time-domain temperature and displacement, thereby improving computational efficiency.

[0029] Preferably, the step of calculating the transient thermal stress of the packaging system according to the time domain temperature and displacement results specifically includes:

[0030] In the first step, the strain is calculated according to the strain-displacement relationship based on the principle of linear thermal expansion;

[0031] In the second step, the transient thermal stress is calculated based on the strain results calculated in the first step and the constitutive relationship between stress and strain.

[0032] Compared with the prior art, the present invention has the following beneficial effects:

[0033] 1. The present invention eliminates the time term in the transient heat conduction equation and the transient stress equation, converts the transient solution process from the time-consuming time stepping form to the efficient order stepping form, and significantly improves the calculation efficiency;

[0034] 2. The present invention does not need to consider the influence of the time step when solving the transient heat conduction equation and the transient stress equation, and has the advantage of unconditional stability of the algorithm. BRIEF DESCRIPTION OF THE DRAWINGS

[0035] Other features, objects and advantages of the present invention will become more apparent from the detailed description of non-limiting embodiments made with reference to the following drawings:

[0036] Figure 1 A flowchart of a numerical simulation method for analyzing transient thermal stress of an electronic packaging system according to the present invention;

[0037] Figure 2 A geometric model diagram of an electronic packaging system according to a specific embodiment;

[0038] Figure 3 A graph showing the change of observation point temperature over time in a specific embodiment;

[0039] Figure 4 FIG. 1 is a graph showing stress variation at an observation point over time for a specific embodiment. DETAILED DESCRIPTION

[0040] The present invention is described in detail below in conjunction with specific embodiments. The following embodiments will help those skilled in the art to further understand the present invention, but are not intended to limit the present invention in any form. It should be noted that, for those of ordinary skill in the art, several changes and improvements can also be made without departing from the concept of the present invention. These all belong to the protection scope of the present invention.

[0041] Specifically, the present invention provides a numerical simulation method for analyzing transient thermal stress of an electronic packaging system, such as Figure 1 As shown, the method comprises the following steps:

[0042] S1: Based on Laguerre polynomial basis functions, the temperature field equation and boundary conditions of the Laguerre domain are established;

[0043] Specifically, in step S1, the steps of establishing the equation are as follows:

[0044] Step S11, expanding the temperature variable in the time domain using a Laguerre basis function;

[0045] Specifically, in step S11, the Laguerre basis function used is:

[0046]

[0047] in, is the time, s is the time scale factor, The expression is:

[0048]

[0049] in, is a Laguerre polynomial of order p.

[0050] The temperature variable T is expanded using the Laguerre basis function shown in expression (1) as follows:

[0051]

[0052] Among them, T p is the temperature Laguerre coefficient.

[0053] Step S12, substituting the expression (3) obtained in step S11 into the time domain temperature field equation, and using the orthogonality of the Laguerre basis function to eliminate the time term, to obtain the temperature field governing equation with the temperature Laguerre coefficient as the unknown quantity;

[0054] Specifically, in step S12, the transient heat conduction equation satisfied by the packaging system is:

[0055]

[0056] Where κ is the thermal conductivity of the material, ρ is the density of the material, c is the heat capacity of the material, r is the spatial position, and g is the time-varying heat source.

[0057] Substitute the expression (3) obtained in step S11 into the time domain temperature field equation (4), and use the orthogonality of the Laguerre basis function to eliminate the time variable, and obtain the heat conduction equation in the Laguerre domain:

[0058]

[0059] Among them, Q q It is the heat source of the Laguerre domain.

[0060] Step S13, substituting the expression (3) obtained in step S11 into the temperature field boundary condition, and using the orthogonality of the Laguerre basis function to eliminate the time term, to obtain the temperature field boundary condition with the temperature Laguerre coefficient as the unknown quantity;

[0061] Specifically, in step S13, the temperature field boundary conditions are as follows:

[0062]

[0063] Among them, Γ 1 is the constant temperature boundary, Γ 2 is the heat flow boundary, Γ 3 is the convection boundary, h is the convection coefficient, q is the heat flux density loaded on the heat flux boundary, T amb is the external ambient temperature, is the gradient along the outer normal direction of Γ.

[0064] Substituting the expression (3) obtained in step S11 into the temperature field boundary condition (6), the temperature field boundary condition of the Laguerre domain is obtained as follows:

[0065] (1) Constant temperature boundary conditions:

[0066]

[0067] in

[0068] (2) Heat flow boundary conditions:

[0069]

[0070] in

[0071] (3) Convective boundary conditions:

[0072]

[0073] in

[0074] S2: Based on the Laguerre polynomial basis function, the stress field equation and boundary conditions of the Laguerre domain are established;

[0075] Specifically, in step S2, the steps of establishing the equation are as follows:

[0076] Step S21, the displacement variable u i In the time domain, it is expanded using the Laguerre basis function:

[0077]

[0078] Among them, u i,p is the displacement Laguerre coefficient.

[0079] Step S22, substituting the expression (10) obtained in step S21 into the time domain stress field equation, and using the orthogonality of the Laguerre basis function to eliminate the time term, to obtain the stress field governing equation with the displacement Laguerre coefficient as the unknown quantity;

[0080] Specifically, in step S22, the momentum conservation equation satisfied by the packaging system is:

[0081]

[0082] Among them, i, j, k are the three directions in the Cartesian coordinate system, μ is the damping coefficient of the material, and u i is the displacement component, σ i and τ ij is the stress tensor component, f i is the time-varying component of the body load.

[0083] Substitute the expression (10) obtained in step S21 into the time domain stress field equation (11), and use the orthogonality of the Laguerre basis function to eliminate the time variable, and obtain the stress equation in the Laguerre domain:

[0084]

[0085] Among them, σ i,q and τ ij,q is the stress tensor component of the Laguerre domain, f i,q is the body load component of the Laguerre domain.

[0086] Step S23, substitute the expression (10) obtained in step S21 into the stress field boundary condition, and use the orthogonality of the Laguerre basis function to eliminate the time term, so as to obtain the stress field boundary condition with the displacement Laguerre coefficient as the unknown quantity.

[0087] Specifically, in step S23, the stress field boundary condition includes a fixed displacement boundary condition.

[0088]

[0089] Among them, Γ u is a fixed displacement boundary, is a fixed displacement vector.

[0090] Substitute the expression (10) obtained in step S21 into the stress field boundary condition (13) to obtain the stress field boundary condition of the Laguerre domain:

[0091]

[0092] S3: Set the material properties, heat sources and boundary conditions of the packaging system, and establish the Laguerre coefficient matrix equations of temperature and displacement based on the finite element method;

[0093] Specifically, in step S3, the steps of establishing the matrix equation are as follows:

[0094] Step S31, setting material properties, heat sources, temperature field boundary conditions and stress field boundary conditions of the packaging system;

[0095] Specifically, in step S31, the thermal properties of the packaging system material include specific heat capacity, density and thermal conductivity, and the mechanical properties include Young's modulus, Poisson's ratio and thermal expansion coefficient; the heat source is the body heat source density loaded into the chip.

[0096] Step S32, combining the parameter setting of step S31, using the finite element method to discretize the space, and sorting out the matrix equation of the temperature Laguerre coefficient;

[0097] Specifically, in step S32, the finite element method is used to discretize the three-dimensional packaging system based on the unstructured grid. In each unit, the q-order Laguerre coefficient of the temperature field can be discretized as:

[0098]

[0099] Among them, N e is the shape function of the element, n is the number of unknowns in the element, is the q-order Laguerre coefficient of the temperature variable at the element node.

[0100] Assemble all units and get the following matrix equation with the following order step

[0101] K T T q =b q +Q q (16)

[0102] Among them, K T is the coefficient matrix in the improved Laguerre domain, b q is the vector associated with the boundary, Q q is the vector consisting of the heat source and the Laguerre coefficients of known temperature.

[0103] Step S33, combining the parameter setting of step S31, using the finite element method to discretize the space, and sorting out the matrix equation of the displacement Laguerre coefficient.

[0104] Specifically, in step S33, the finite element method is used to discretize the three-dimensional packaging system based on the unstructured grid. In each unit, the q-order Laguerre coefficient of the stress field can be discretized as:

[0105]

[0106] in is the q-order Laguerre coefficient of the displacement vector component at the element node.

[0107] Assemble all units and get the following matrix equation with the following order step

[0108] (M F +D F +K F ) q =f q (18)

[0109] Among them, M T , D T and K T are coefficient matrices in the improved Laguerre domain, f q is a vector consisting of the known temperature Laguerre coefficients and the known displacement Laguerre coefficients.

[0110] S4: Solve the matrix equations step by step to obtain the Laguerre coefficients of temperature and displacement in turn;

[0111] Specifically, in step S4, the step-by-step solution refers to gradually increasing the order of the Laguerre basis function from the 0th order to obtain the corresponding matrix equations (16) and (18), and using a large sparse matrix solver to solve the equations to obtain Laguerre coefficients of each order.

[0112] S5: Based on the Laguerre coefficient, the time domain temperature and displacement distribution of the packaging system are reconstructed;

[0113] Specifically, in step S5, the steps of reconstructing the result are as follows:

[0114] Step S51, based on the solved temperature and displacement Laguerre coefficients and in combination with the Laguerre basis function expansion formula, reconstruct the time domain temperature and displacement distribution of the packaging system;

[0115] Specifically, in step S51, the reconstruction formula is:

[0116]

[0117] Where N is the Laguerre order required for actual calculation.

[0118] Step S52, using a data filtering method to reduce the Laguerre order required for reconstructing the time domain temperature and displacement, thereby improving the calculation efficiency.

[0119] Specifically, in step S52, the data filtering formula used is:

[0120] v f (N c )=2av f (N c -1)-a 2 v f (N c -2)+(1-a) 2 v(N c ) (twenty one)

[0121] Among them, a is the filter coefficient with a value between 0 and 1, and v f is the result after data filtering.

[0122] S6: Based on the time domain temperature and displacement results, the transient thermal stress of the packaging system is calculated.

[0123] Specifically, in step S6, the calculation steps are as follows:

[0124] Step S61, calculating the strain according to the strain-displacement relationship and in combination with the linear thermal expansion principle;

[0125] Specifically, in step S61, the calculation formula is:

[0126]

[0127] Among them, ε i and γ ij are the components of the strain tensor.

[0128] Step S62, calculating transient thermal stress according to the strain result calculated in step S61 and combining the constitutive relationship between stress and strain.

[0129] Specifically, in step S62, the calculation formula is:

[0130]

[0131] Where α is the coefficient of thermal expansion, and the Lame constant λ and G are calculated from Young's modulus E and Poisson's ratio ν:

[0132]

[0133] According to the above simulation method, a specific embodiment is calculated.

[0134] In this embodiment, the geometric model of the electronic packaging system is as follows: Figure 2 As shown, observation point P1 is set at the center of the chip and P2 is set at the surface of the PCB copper layer. The outer surface of the electronic packaging system is the convection boundary, and the convection heat transfer coefficient is 10W / (m 2 K), and the external ambient temperature is 293.15 K. The lower surface of the packaging system is set to a fixed displacement boundary condition with a displacement of (0 mm, 0 mm, 0 mm).

[0135] The calculations were performed using COMSOL Multiphysics software and the method of the present invention, respectively, and the time interval for transient analysis was 60 seconds. The temperature response of the observation point over time is shown in FIG. Figure 3 As shown in FIG. 1 , the maximum absolute errors of COMSOL Multiphysics software and the method of the present invention are 0.0079 K and 0.07 K, respectively; the stress response of the observation point changing with time is shown in FIG. Figure 4 As shown, the maximum absolute errors are 0.0062MPa and 0.037MPa respectively. The above comparison results fully illustrate the accuracy of the method of the present invention.

[0136] In terms of analysis efficiency, the simulation time of COMSOL Multiphysics software is 10 minutes and 15 seconds, while the simulation time of the method of the present invention is 27 seconds. Compared with COMSOL Multiphysics software, the efficiency of the method of the present invention is improved by 22 times, which shows the efficiency advantage of the method of the present invention.

[0137] The above describes the specific embodiments of the present invention. It should be understood that the present invention is not limited to the above specific embodiments, and those skilled in the art can make various changes or modifications within the scope of the claims, which does not affect the essence of the present invention. In the absence of conflict, the embodiments of the present application and the features in the embodiments can be combined with each other arbitrarily.

Claims

1. A numerical simulation method for analyzing transient thermal stress of electronic packaging systems. It is characterized in that The method comprises the following steps: Based on Laguerre polynomial basis functions, the temperature field equation and boundary conditions of the Laguerre domain are established; Based on Laguerre polynomial basis functions, the stress field equations and boundary conditions of the Laguerre domain are established; Set the material properties, heat sources and boundary conditions of the packaging system, and establish the Laguerre coefficient matrix equations of temperature and displacement based on the finite element method; Solve the matrix equations step by step to obtain the Laguerre coefficients of temperature and displacement in turn; Based on the Laguerre coefficient, the time domain temperature and displacement distribution of the packaging system are reconstructed; Based on the time-domain temperature and displacement results, the transient thermal stress of the packaging system is calculated.

2. The numerical simulation method for analyzing transient thermal stress of an electronic packaging system according to claim 1, It is characterized in that The step of establishing the temperature field equation and boundary conditions of the Laguerre domain based on the Laguerre polynomial basis function specifically includes: In the first step, the temperature variable is expanded in the time domain using the Laguerre basis function; The second step is to substitute the expression obtained in the first step into the time-domain temperature field equation, and use the orthogonality of the Laguerre basis function to eliminate the time term, and obtain the temperature field governing equation with the temperature Laguerre coefficient as the unknown quantity; In the third step, the expression obtained in the first step is substituted into the temperature field boundary condition, and the time term is eliminated by using the orthogonality of the Laguerre basis function to obtain the temperature field boundary condition with the temperature Laguerre coefficient as the unknown quantity.

3. The numerical simulation method for analyzing transient thermal stress of an electronic packaging system according to claim 1, It is characterized in that The step of establishing the stress field equation and boundary conditions of the Laguerre domain based on the Laguerre polynomial basis function specifically includes: In the first step, the displacement variables are expanded in the time domain using Laguerre basis functions; In the second step, the expression obtained in the first step is substituted into the stress field equation in the time domain, and the time term is eliminated by using the orthogonality of the Laguerre basis function to obtain the stress field governing equation with the displacement Laguerre coefficient as the unknown quantity; In the third step, the expression obtained in the first step is substituted into the stress field boundary condition, and the time term is eliminated by using the orthogonality of the Laguerre basis function to obtain the stress field boundary condition with the displacement Laguerre coefficient as the unknown quantity.

4. The numerical simulation method for analyzing transient thermal stress of an electronic packaging system according to claim 1, It is characterized in that The steps of setting the material properties, heat sources and boundary conditions of the packaging system and establishing the Laguerre coefficient matrix equation of temperature and displacement based on the finite element method specifically include: The first step is to set the material properties, heat source, temperature field boundary conditions and stress field boundary conditions of the packaging system; In the second step, combined with the parameter setting in the first step, the finite element method is used to discretize the space and the matrix equation of the temperature Laguerre coefficient is obtained; In the third step, combined with the parameter settings in the first step, the finite element method is used to discretize the space and the matrix equation of the displacement Laguerre coefficient is obtained.

5. The numerical simulation method for analyzing transient thermal stress of an electronic packaging system according to claim 1, It is characterized in that In the step of solving the matrix equations step by step to obtain the Laguerre coefficients of temperature and displacement in turn, the step-by-step solution refers to increasing the order of the Laguerre basis function from the 0th order to obtain the corresponding matrix equation, and using a large sparse matrix solver to solve the equation group to obtain the Laguerre coefficients of each order.

6. The numerical simulation method for analyzing transient thermal stress of an electronic packaging system according to claim 1, It is characterized in that The step of reconstructing the time domain temperature and displacement distribution of the packaging system based on the Laguerre coefficient specifically includes: In the first step, based on the obtained temperature and displacement Laguerre coefficients and combined with the Laguerre basis function expansion formula, the time domain temperature and displacement distribution of the packaging system are reconstructed; In the second step, data filtering is used to reduce the Laguerre order required to reconstruct the time-domain temperature and displacement, thereby improving computational efficiency.

7. The numerical simulation method for analyzing transient thermal stress of an electronic packaging system according to claim 1, It is characterized in that The step of calculating the transient thermal stress of the packaging system according to the time domain temperature and displacement results specifically includes: In the first step, the strain is calculated according to the strain-displacement relationship based on the principle of linear thermal expansion; In the second step, the transient thermal stress is calculated based on the strain results calculated in the first step and the constitutive relationship between stress and strain.

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