A numerical simulation method for analyzing transient thermal stress in electronic packaging systems

Abstract

This invention provides a numerical simulation method for analyzing transient thermal stress in electronic packaging systems, comprising: establishing the temperature field equation and boundary conditions of the Laguerre domain based on Laguerre polynomial basis functions; establishing the stress field equation and boundary conditions of the Laguerre domain based on Laguerre polynomial basis functions; setting the material properties, heat sources, and boundary conditions of the packaging system, and establishing the Laguerre coefficient matrix equations for temperature and displacement respectively based on the finite element method; solving the matrix equations order by order to obtain the Laguerre coefficients of temperature and displacement; reconstructing the time-domain temperature and displacement distribution of the packaging system based on the Laguerre coefficients; and calculating the transient thermal stress of the packaging system based on the time-domain temperature and displacement results. This invention, while ensuring accuracy, has the advantages of unconditional stability and high efficiency.

Description

Technical Field

[0001] This invention relates to the field of numerical simulation calculation technology, and more specifically, to a numerical simulation method for analyzing transient thermal stress in electronic packaging systems. Background Art

[0002] The ever-increasing power consumption of electronic devices poses a severe challenge to the thermal management of electronic packaging systems. Simultaneously, high temperatures and mismatches in mechanical parameters between different materials significantly reduce the stability and shorten the lifespan of packaging systems. Therefore, accurately and efficiently obtaining the temperature and stress distribution of packaging systems is crucial for engineers in the early design stages.

[0003] Compared to costly experimental methods, numerical simulation can conveniently and intuitively provide detailed information on temperature and stress in 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. A small time step, while ensuring algorithm stability, leads to excessively long simulation times; conversely, a large time step can effectively shorten simulation time, but typically faces serious problems such as deterioration in computational accuracy and loss of algorithm stability. Therefore, for transient thermal stress analysis scenarios in packaging systems, it is necessary to develop an efficient numerical simulation method to address these bottlenecks. Summary of the Invention

[0004] To address the shortcomings of existing technologies, the present invention aims to provide a numerical simulation method for analyzing transient thermal stress in electronic packaging systems. This method, while ensuring accuracy, has the advantages of unconditional stability and high efficiency.

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

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

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

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

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

[0010] By solving the matrix equations step by step, the Laguerre coefficients of temperature and displacement are obtained in turn.

[0011] Based on the Laguerre coefficient, the time-domain temperature and displacement distribution of the packaging system were 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 equations and boundary conditions of the Laguerre domain based on Laguerre polynomial basis functions specifically includes:

[0014] The first step is to expand the temperature variable in the time domain using the Laguerre 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 functions to eliminate the time term, so as to obtain the temperature field dominance equation with the temperature Laguerre coefficient as the unknown.

[0016] The third step is to substitute the expression obtained in the first step into the temperature field boundary conditions, and use the orthogonality of the Laguerre basis functions to eliminate the time term, so as to obtain the temperature field boundary conditions with the temperature Laguerre coefficient as the unknown.

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

[0018] The first step is to expand the displacement variable in the time domain using the Laguerre basis function;

[0019] The second step is to substitute the expression obtained in the first step into the time-domain stress field equation and use the orthogonality of the Laguerre basis functions to eliminate the time term, so as to obtain the stress field dominance equation with the displacement Laguerre coefficient as the unknown.

[0020] The third step is to substitute the expression obtained in the first step into the stress field boundary conditions, and use the orthogonality of the Laguerre basis functions to eliminate the time term, so as to obtain the stress field boundary conditions with the displacement Laguerre coefficient as the unknown.

[0021] Preferably, the step of setting the material properties, heat source, and boundary conditions of the encapsulation system, and establishing the Laguerre coefficient matrix equations for 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] The second step involves combining the parameter settings from the first step with the finite element method to discretize the space and then assemble the matrix equation for the temperature Laguerre coefficient.

[0024] The third step involves combining the parameter settings from the first step with the finite element method to discretize the space and then rearrange the matrix equations for the displacement Laguerre coefficients.

[0025] Preferably, in the step of solving the matrix equations step by step to obtain the Laguerre coefficients of temperature and displacement, the step-by-step solution refers to successively increasing the order of the Laguerre basis function from the 0th order to obtain the corresponding matrix equations, and using a large sparse matrix solver to solve the system of equations 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] The first step is to reconstruct the time-domain temperature and displacement distribution of the encapsulation system based on the obtained temperature and displacement Laguer coefficients and the Laguer basis function expansion formula.

[0028] The second step involves using data filtering 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 based on the time-domain temperature and displacement results specifically includes:

[0030] The first step is to calculate the strain based on the principle of linear thermal expansion and the strain-displacement relationship.

[0031] The second step is to calculate the transient thermal stress based on the strain results obtained in the first step, combined with the constitutive relationship between stress and strain.

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

[0033] 1. This invention eliminates the time terms in the transient heat conduction equation and transient stress equation, transforming the transient solution process from a time-consuming time-stepping form to an efficient order-stepping form, significantly improving computational efficiency;

[0034] 2. This invention does not require consideration of the time step when solving the transient heat conduction equation and transient stress equation, and has the advantage of unconditional algorithm stability. Attached Figure Description

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

[0036] Figure 1 The flowchart of the numerical simulation method for analyzing transient thermal stress in electronic packaging systems according to this invention is shown below.

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

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

[0039] Figure 4 The graph shows the stress-time variation at an observation point in a specific embodiment. Detailed Implementation

[0040] The present invention will now be described in detail with reference to specific embodiments. These embodiments will help those skilled in the art to further understand the present invention, but do not limit the invention in any way. It should be noted that those skilled in the art can make several changes and improvements without departing from the concept of the present invention. These all fall within the protection scope of the present invention.

[0041] Specifically, this invention provides a numerical simulation method for analyzing transient thermal stress in electronic packaging systems, such as... Figure 1 As shown, the method includes the following steps:

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

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

[0044] Step S11: Expand the temperature variable in the time domain using the Laguerre function;

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

[0046]

[0047] in, It represents time, where 's' is the time scale factor. The expression is:

[0048]

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

[0050] Expanding the temperature variable T using the Laguerre function shown in expression (1) yields:

[0051]

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

[0053] Step S12, substitute the expression (3) obtained in step S11 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 dominance equation with the temperature Laguerre coefficient as the unknown.

[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 location, and g is the heat source that changes with time.

[0057] Substituting the expression (3) obtained in step S11 into the time-domain temperature field equation (4), and using the orthogonality of the Laguerre basis functions to eliminate the time variable, we 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: Substitute the expression (3) obtained in step S11 into the temperature field boundary conditions, and use the orthogonality of the Laguerre basis function to eliminate the time term, so as to obtain the temperature field boundary conditions with the temperature Laguerre coefficient as the unknown.

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

[0062]

[0063] Where Γ1 is the isothermal boundary, Γ2 is the heat flux boundary, Γ3 is the convective boundary, h is the convective coefficient, q is the heat flux density applied to the heat flux boundary, and T is the heat flux density applied to the heat flux boundary. amb It is the external ambient temperature. It is the gradient along the outward 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) Isothermal boundary conditions:

[0066]

[0067] in

[0068] (2) Heat flow boundary conditions:

[0069]

[0070] in

[0071] (3) Convection boundary conditions:

[0072]

[0073] in

[0074] S2: Based on the Laguerre polynomial basis functions, establish the stress field equations and boundary conditions for the Laguerre domain;

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

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

[0077]

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

[0079] Step S22: Substitute the expression (10) obtained in step S21 into the time-domain stress field equation, and use the orthogonality of the Laguerre basis function to eliminate the time term, so as to obtain the stress field dominance equation with the displacement Laguer coefficient as the unknown.

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

[0081]

[0082] Where i, j, k are three directions in the Cartesian coordinate system, μ is the damping coefficient of the material, and u i It is the displacement component, σ i and τ ij It is the stress tensor component, f i It is the body load component that varies with time.

[0083] Substituting the expression (10) obtained in step S21 into the time-domain stress field equation (11), and using the orthogonality of the Laguerre basis functions to eliminate the time variable, we obtain the stress equation in the Laguerre domain:

[0084]

[0085] Where, σ i,q and τ ij,q It is the stress tensor component of the Laguerre domain, f i,q It is the volume load component of the Laguerre domain.

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

[0087] Specifically, in step S23, the stress field boundary conditions include fixed displacement boundary conditions.

[0088]

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

[0090] Substituting the expression (10) obtained in step S21 into the stress field boundary condition (13), we obtain the stress field boundary condition for the Laguerre domain:

[0091]

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

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

[0094] Step S31: Set the material properties, heat source, 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 coefficient of thermal expansion; the heat source is the volume heat source density loaded onto the chip.

[0096] Step S32: Combining the parameter settings of step S31, the space is discretized using the finite element method, and the matrix equation of the temperature Laguerre coefficient is obtained.

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

[0098]

[0099] Where, N e is the shape function of the element, and n is the number of unknowns within the element. It is the q-th order Laguerre coefficient of the temperature variable at the unit node.

[0100] Assemble all the units to obtain the following matrix equation with a step order.

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

[0102] Among them, K T For the coefficient matrix in the improved Laguerre field, b q Q is a vector related to the boundary. q It is a vector consisting of the heat source, the known temperature, and the Laguerre coefficient.

[0103] In step S33, based on the parameter settings in step S31, the space is discretized using the finite element method, and the matrix equation of the displacement Laguerre coefficients is obtained.

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

[0105]

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

[0107] Assemble all the units to obtain the following matrix equation with a step order.

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

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

[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 stepwise solution refers to successively 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 equation set to obtain the 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 for obtaining the reconstruction result are as follows:

[0114] Step S51: Based on the obtained temperature and displacement Laguerre coefficients, and combined 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 the actual calculation.

[0118] Step S52: Data filtering is used to reduce the Laguerre order required to reconstruct the time-domain temperature and displacement, thereby improving computational 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] Where a is a filter coefficient with a value between 0 and 1, v f This is the result after data filtering.

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

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

[0124] Step S61: Calculate the strain based on the strain-displacement relationship and in conjunction with the principle of linear thermal expansion;

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

[0126]

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

[0128] Step S62: Based on the strain results obtained in step S61, and combined with the constitutive relationship between stress and strain, calculate the transient thermal stress.

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

[0130]

[0131] Where α represents the coefficient of thermal expansion, and Lamé constants λ and G are calculated using Young's modulus E and Poisson's ratio ν:

[0132]

[0133] Based on 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 points P1 are located at the center of the chip, and P2 is located on the surface of the PCB copper layer. The outer surface of the electronic packaging system is a convective boundary with a convective heat transfer coefficient of 10 W / (m²). 2 The 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] Calculations were performed using COMSOL Multiphysics software and the method of this invention, respectively, with a transient analysis time interval of 60 seconds. The temperature response at the observation point over time is as follows: Figure 3 As shown, the maximum absolute errors of COMSOL Multiphysics software and the method of this invention are 0.0079K and 0.07K, respectively; the stress response at the observation point over time is as follows: Figure 4 As shown, the maximum absolute errors are 0.0062 MPa and 0.037 MPa, respectively. The above comparison results fully demonstrate 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 this invention is 27 seconds. Compared with COMSOL Multiphysics software, the efficiency of the method of this invention is improved by 22 times, demonstrating the efficiency advantage of the method of this invention.

[0137] Specific embodiments of the present invention have been described above. It should be understood that the present invention is not limited to the specific embodiments described above, and those skilled in the art can make various changes or modifications within the scope of the claims, which do not affect the essence of the present invention. Unless otherwise specified, the embodiments and features described in this application can be arbitrarily combined with each other.

Claims

1. A numerical simulation method for analyzing transient thermal stress in an electronic packaging system, characterized in that, The method includes the following steps: For the transient heat conduction equation satisfied by the packaging system: 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 location, and g is the heat source that varies with time. Based on the Laguerre polynomial basis functions, the temperature field equation and boundary conditions of the Laguerre domain are established. For the momentum conservation equation satisfied by the packaging system: Where i, j, k are three directions in the Cartesian coordinate system, μ is the damping coefficient of the material, and u i It is the displacement component, σ i and τ ij It is the stress tensor component, f i These are time-varying body load components. Based on Laguerre polynomial basis functions, the stress field equations and boundary conditions for the Laguerre domain are established, specifically including: The first step is to change the displacement variable u. i In the time domain, expanded using the Laguerre function, it is: Among them, u i,p It is the Laguerre coefficient for displacement; The second step involves substituting the expression obtained in the first step into the time-domain stress field equation, and using the orthogonality of the Laguerre basis functions to eliminate the time term, resulting in the stress field governing equation with the displacement Laguer coefficient as the unknown: Where, σ i,q and τ ij,q It is the stress tensor component of the Laguerre domain, f i,q It is the volume load component of the Laguerre domain; The third step is to substitute the expression obtained in the first step into the stress field boundary conditions, and use the orthogonality of the Laguerre basis functions to eliminate the time term, thus obtaining the stress field boundary conditions with the displacement Laguer coefficient as the unknown: Set the material properties, heat source and boundary conditions of the packaging system, and establish the Laguerre coefficient matrix equations for temperature and displacement based on the finite element method; By solving the matrix equations step by step, the Laguerre coefficients of temperature and displacement are obtained in turn. Based on the Laguerre coefficient, the time-domain temperature and displacement distribution of the packaging system were 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 in an electronic packaging system according to claim 1, characterized in that, The steps for establishing the temperature field equations and boundary conditions in the Laguerre domain based on Laguerre polynomial basis functions specifically include: The first step is to expand the temperature variable in the time domain using the Laguerre 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 functions to eliminate the time term, so as to obtain the temperature field dominance equation with the temperature Laguerre coefficient as the unknown. The third step is to substitute the expression obtained in the first step into the temperature field boundary conditions, and use the orthogonality of the Laguerre basis functions to eliminate the time term, so as to obtain the temperature field boundary conditions with the temperature Laguerre coefficient as the unknown.

3. The numerical simulation method for analyzing transient thermal stress in an electronic packaging system according to claim 1, characterized in that, The steps of setting the material properties, heat source, and boundary conditions of the encapsulation system, and establishing the Laguerre coefficient matrix equations for 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; The second step involves combining the parameter settings from the first step with the finite element method to discretize the space and then assemble the matrix equation for the temperature Laguerre coefficient. The third step involves combining the parameter settings from the first step with the finite element method to discretize the space and then rearrange the matrix equations for the displacement Laguerre coefficients.

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

5. The numerical simulation method for analyzing transient thermal stress in an electronic packaging system according to claim 1, 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: The first step is to reconstruct the time-domain temperature and displacement distribution of the encapsulation system based on the obtained temperature and displacement Laguer coefficients and the Laguer basis function expansion formula. The second step involves using data filtering to reduce the Laguerre order required to reconstruct the time-domain temperature and displacement, thereby improving computational efficiency.

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

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