CN111737835A - Three-period minimum curved surface-based three-dimensional porous heat dissipation structure design and optimization method - Google Patents

Three-period minimum curved surface-based three-dimensional porous heat dissipation structure design and optimization method Download PDF

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CN111737835A
CN111737835A CN202010597487.1A CN202010597487A CN111737835A CN 111737835 A CN111737835 A CN 111737835A CN 202010597487 A CN202010597487 A CN 202010597487A CN 111737835 A CN111737835 A CN 111737835A
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王胜法
姜钰
李宝军
王祎
李凤岐
罗钟铉
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Abstract

A three-cycle minimum curved surface-based design and optimization method for a three-dimensional porous heat dissipation structure belongs to the field of computer aided design. First, a porous structure is built by an implicit function representation of a three-cycle infinitesimal surface. Secondly, according to a steady-state heat conduction equation, the heat dissipation problem is converted into a minimization problem of heat dissipation weakness under a given constraint condition. Then, a global-local interpolation method is used for directly solving the parameter function. And finally, carrying out periodic optimization and wall thickness optimization on the modeling problem to obtain the optimized porous sheath-like structure with smooth period and wall thickness variation. The porous structure of the present invention greatly improves heat dissipation performance, heat conduction efficiency and efficiency. The porous structure designed by the invention has the characteristics of smoothness, full connectivity, controllability, quasi-self-supporting property and the like, the applicability and manufacturability of the structure are ensured by the characteristics, the porous structure is suitable for a 3D printing manufacturing method, the internal structure in the printing process does not need to be additionally supported, and the printing time and the printing material are saved.

Description

Three-period minimum curved surface-based three-dimensional porous heat dissipation structure design and optimization method
Technical Field
The invention belongs to the field of engineering design and manufacture, and relates to a design and optimization method of a three-dimensional porous heat dissipation structure, which is suitable for heat dissipation structures of various large-scale engineering machinery, radiators of automobiles and the like and related parts of combustion appliances.
Background
How to construct a light and efficient heat dissipation structure has received wide attention in various engineering fields. The conventional radiator structure cannot achieve high heat conduction efficiency. Through the research on the porous structure, the heat-conducting property of the structure can be effectively improved, however, the representation and optimization method becomes a technical bottleneck restricting the further development of the structure.
The traditional heat dissipation structure design depends on the basic theory and practical experience of calorifics, and the heat dissipation problem of a complex structure is difficult to solve. Subsequently, topology optimization methods using porous structures such as tree topologies, truss/frame structures, microstructures, etc. have emerged to address the above issues, and these porous structures can be used to calculate high-degree-of-freedom topologies for cooling channels. However, these methods have a common problem in that they require a large number of design variables and are expensive to optimize due to time-consuming re-gridding.
In recent years, porous structures based on three-cycle extremely small curved surfaces have been widely used in the fields of tissue engineering, lightweight manufacturing, biomedicine, and the like. The porous structure based on the three-cycle extremely-small curved surface has the advantages of good connectivity, easiness in control, high specific strength, rigidity and the like. The porous structure based on the three-cycle infinitesimal surface (especially for large and complex porous structures) is very time-consuming and memory-consuming to represent by a polyhedral mesh (tetrahedron or hexahedron), and the traditional finite element-based porous structure processing method is almost heuristic without effective optimization, so that the porous structure based on the three-cycle infinitesimal surface is less researched in heat dissipation.
In view of the above, an efficient representation and optimization method is proposed to obtain the period and wall thickness of a three-cycle minimum curved porous sheath structure suitable for heat dissipation. The main optimization processes include period optimization and wall thickness optimization. The former is a coarse adjustment of the structure and the latter is a fine adjustment of the structure. Firstly, expressing a porous sheath-like structure by using an implicit function, wherein the implicit function is controlled by a periodic parameter function and a wall thickness parameter function; on the basis, a steady-state heat conduction equation with boundary conditions can be conveniently and directly established into a mathematical model by using a function expression, and then the optimization problem of the model is converted into the solution of the two continuous parameter functions; and finally, from the perspective of discretization, the two parameter functions can be effectively calculated by utilizing implicit function representation and radial basis interpolation without re-meshing, and finally the optimized heat dissipation porous structure with smooth period and wall thickness variation is obtained.
Disclosure of Invention
The invention provides an effective expression and optimization method of a porous heat dissipation structure based on a three-cycle extremely-small curved surface. First, a porous structure is built by an implicit function representation of a three-cycle infinitesimal surface. Secondly, according to a steady-state heat conduction equation, the heat dissipation problem is converted into a minimization problem of heat dissipation weakness under a given constraint condition. Then, a global-local interpolation method is used for directly solving the parameter function. Finally, we perform periodic optimization and wall thickness optimization on the modeling problem to obtain an optimized porous sheath-like structure with smooth period and wall thickness variation.
The technical scheme adopted by the invention is as follows:
a three-cycle extremely-small-curved-surface-based design and optimization method for a three-dimensional porous heat dissipation structure comprises the following steps:
method for expressing porous structure
The commonly used three-cycle minimum curved surface mostly has implicit function representation, taking the P minimum curved surface as an example:
Figure BDA0002557908090000021
wherein r is a three-dimensional vector, and x, y and z are corresponding coordinates of the three-dimensional vector.
The periodic parameter function P (r) >0 can be directly added into the function representation of the three-period extremely-small curved surface, and in order to make the distance scale of the directional distance field in the periodic variation process basically unchanged, the implicit function representation is improved as follows:
Figure BDA0002557908090000022
wherein P (r) controls the continuous change of the pore period, and a porous curved surface with smooth transition on the space is constructed; other types of three-cycle pole faces were treated in the same manner.
The porous structure with thickness based on the three-cycle extremely-small curved surface can be used for the improved implicit function curved surface by controlling the parameter function W (r) of the wall thickness
Figure BDA0002557908090000023
Two offset surfaces are obtained after the offset to two sides, and are expressed as:
Figure BDA0002557908090000024
Figure BDA0002557908090000025
finally, the three-cycle minimal surface-based porous sheath-like structure is represented by an intersection operator continuous function:
Figure BDA0002557908090000031
in the above definition, the parameter functions p (r) >0 for controlling the period and the parameter functions w (r) >0 for controlling the wall thickness are introduced to realize the control of the shape and the periodic pores of the porous structure, and the porous structure with the wall thickness meeting the requirement is finally generated by optimizing the parameter functions p (r) and w (r).
The porous structure defined by the function inherits the excellent characteristics of the three-period extremely-small curved surface, such as high surface area-volume ratio, full connectivity, high smoothness and controllability. High surface to volume ratio and full connectivity facilitate heat dissipation from the structure. The structural function provides a computationally optimized method based on the high degree of controllability of the period and wall thickness. Good smoothness and connectivity facilitates 3D printing manufacturing, ensures manufacturing accuracy, and can remove excess material (e.g., excess liquid in SLA) in 3D manufacturing.
(II) optimization procedure for heat dissipation problem
The invention mainly focuses on the heat dissipation problem under the condition of steady-state heat conduction, after a model heat source and boundary conditions are given, the internal space of the model is filled by using the constructed porous sheath-like structure, and the optimal distribution of the porous structure period and the wall thickness is solved under the conditions of given material volume constraint and periodic function gradient constraint.
1. Problem modeling
Based on the above purpose, the heat dissipation problem model is established as follows:
Figure BDA0002557908090000032
such that:
Figure BDA0002557908090000033
Figure BDA0002557908090000034
Figure BDA0002557908090000035
where C is the heat dissipation weakness, T is the temperature field, Ω is the given design area, Φ is the functional representation of the porous sheath-like structure given above, Q is the heat flux density of the internal heat source, Q is the heat flux density of the internal heat sourcesIs the Neumann boundaryQThe heat flux in the upper direction of the normal line,
Figure BDA0002557908090000036
is a given temperature on the Directilet boundary, λ is the thermal conductivity;
Figure BDA0002557908090000037
is a vector differential operator which is a function of,
Figure BDA0002557908090000038
x, Y, Z represent unit vectors in the positive direction of the three coordinate axes x, y, and z, respectively;
Figure BDA0002557908090000039
is the corresponding test function and is,
Figure BDA00025579080900000310
Sob1is a first order Sobelov space, V is the volume of the porous structure,
Figure BDA0002557908090000041
is corresponding volume constraint, and adds gradient constraint of periodic variation to avoid the damage of porous structure caused by severe variation of periodic function
Figure BDA0002557908090000042
And formula for calculation of a mode having a gradient
Figure BDA0002557908090000043
H (x) is the Heaviside function, H (x) is 0 when x is negative, otherwise 1, and H (x) is defined as the continuous function H in order to minimize the optimization problem and avoid checkerboard phenomenaη(x) The function is defined as:
Figure BDA0002557908090000044
wherein η is a regularization parameter for controlling the number of non-zero elements in the global stiffness matrix, which is generally taken as parameter η -10-3An interval of intermediate values is defined. Furthermore, the thermal conductivity λ of the material of the porous structure is calculated from the structure function Φ and should be set as:
Figure BDA0002557908090000045
ξ H (phi) is the volume ratio of the solid material, lambdaSAnd λDRepresenting the thermal conductivity of the solid material and the pore portion, respectively.
2. Discretization
The method is characterized in that a double-scale grid is adopted in the discretization process, the three-dimensional design space of a design domain is divided into uniform hexahedron finite elements called coarse units firstly, the coarse units are used for generating a temperature field, and the number n of the coarse unitssIs determined by the volume of the design space; each coarse cell is then further subdivided into smaller hexahedral cells, called fine cells, which are used for more precise geometric calculations of volume, etc., where the number n of fine cells in each coarse cellbThe default setting is 27. A discrete form of the optimization problem (1.6-1.9) is obtained:
Figure BDA0002557908090000046
such that:
KT＝Q (1.12)
Figure BDA0002557908090000047
Figure BDA0002557908090000048
where T is the temperature field, Q is the heat source and heat flux terms, K is the stiffness matrix, V is the volume of the porous structure,
Figure BDA0002557908090000051
is a corresponding volume constraint, Nb＝nb×nsIs the total number of the thin cells,
Figure BDA0002557908090000052
is the function value of phi of the l-th node in the j-th cell, vbIs the volume of the fine grid cell, G is the global gradient constraint of the structure, | Ω | is the volume of Ω, nlIs the number of sub-regions in the design domain,
Figure BDA0002557908090000053
is the ith sub-region omegaiThe number of the thin units in the inner,
Figure BDA0002557908090000054
is the gradient of the periodic function at the ith point in the s-th cell,
Figure BDA0002557908090000055
is the ith sub-region omegaiThe local gradient within the constraint value of the local gradient,
Figure BDA0002557908090000056
is the volume of the ith coarse grid cell, p>0 is a penalty factor for the global gradient constraint and has:
Figure BDA0002557908090000057
3. global-local interpolation
The optimization of a period parameter function and a thickness parameter function is converted into the optimization of a limited number of design variables by adopting a global-local radial basis interpolation algorithm, and the key idea is to decompose a large coefficient matrix into a weighted small coefficient matrix for solving.
Taking the periodic parameter function as an example, Ω is first divided into nlSub-regions
Figure BDA0002557908090000058
The radial basis interpolation in the local ellipsoid (containing the corresponding sub-region) yields the local periodic parameter function:
Figure BDA0002557908090000059
Figure BDA00025579080900000510
wherein psik(r) is a number fromk(r) defined weight parameter, dk(r)＝‖r-Ck‖2Is interpolated to the ellipsoid central point CkDistance of (1) ()+Is that the truncation function satisfies x>Time 0 (x)+X, otherwise (x)+＝0,Rk(r) is a length function with respect to the radius, Pk(r) is the subregion ΩkA corresponding local periodic parameter function within the local ellipsoid, and defined as:
Figure BDA00025579080900000511
wherein R iski(r)＝(r-Oki)2log(|r-Oki|) is the sheet radial basis function,
Figure BDA00025579080900000512
is the sub-region omegakControl points in the corresponding local ellipsoid, qki(r) is a primary term of coordinates x, y, z, akiAnd bkjThe coefficients to be found are the quadratic term and the primary term, respectively, and m is the number of primary terms (m is 4 by default).
Global-local radial basis interpolation can be simplified to the following form:
Figure BDA00025579080900000513
wherein n istIs the total number of control points in the design field omega (generally 400), Ni(r) is the corresponding calculable coefficient function,
Figure BDA0002557908090000061
is a function of the period of the control point. It proposes to provideThe global local interpolation method can improve the calculation efficiency and simultaneously lead the structure to change smoothly.
4. Modeling problem optimization
The three-dimensional heat dissipation optimization method based on the constructed optimization problem comprises two parts of period optimization and wall thickness optimization. The period and the wall thickness of the porous structure based on the three-period extremely-small curved surface are independently controlled by a period function P (r) and a wall thickness function W (r), the period optimization is coarse adjustment of the structure, the wall thickness optimization is fine adjustment, and the specific optimization process is as follows:
step 1: optimizing the period; firstly, converting function optimization into optimization of interpolation basis function parameters by using a radial basis interpolation method; randomly selecting n in the solution domaintA base point of interpolation
Figure BDA0002557908090000062
Then there is the interpolation form:
Figure BDA0002557908090000063
thus, the optimization problem is transformed into a parametric variable
Figure BDA0002557908090000064
The optimization problem of (2); finally, the optimization variables are derived by the objective function and the constraint function as follows:
Figure BDA0002557908090000065
Figure BDA0002557908090000066
Figure BDA0002557908090000067
Figure BDA0002557908090000068
wherein the content of the first and second substances,
Figure BDA0002557908090000069
the objective function, the volume constraint and the gradient constraint respectively to the parametric variable PiThe equation for the partial derivative is solved,
Figure BDA00025579080900000610
is an intermediate equation to be calculated in the process of gradient partial derivative calculation; n issIs the number of coarse cells, nbIs the number of thin cells in each thick cell;
Figure BDA00025579080900000611
is phi function value of the l node in the k coarse unit and the j fine unit; from the formula of thermal conductivity
Figure BDA00025579080900000612
Nothing ξkjIs the heat conductivity lambda of the k-th coarse cell and the j-th fine cellkjCorresponding parameter factor ξ, K0Is the initial stiffness matrix. In MMA solver, by
Figure BDA0002557908090000071
An optimized porous structure with a smoothly varying period can be obtained. Since the wall thickness function w (r) is fixed, the structural porosity as a whole increases with increasing periodic function p (r), and therefore a period-optimized convergence is easy to achieve. In our experiment, the cycle optimization converged to 70 iterations.
Step 2: optimizing the wall thickness; similarly, the control point based on W (r) (variable is
Figure BDA0002557908090000072
) And constructing a wall thickness function W (r) by adopting a radial basis interpolation method, wherein the corresponding sensitivity analysis is as follows:
Figure BDA0002557908090000073
Figure BDA0002557908090000074
wherein the content of the first and second substances,
Figure BDA0002557908090000075
respectively, an objective function, a volume constraint, and a parametric variable WiSolving an equation of partial derivative;
Figure BDA0002557908090000076
is the phi function value of the ith node in the kth coarse cell and the jth fine cell ξkjIs the heat conductivity lambda of the k-th coarse cell and the j-th fine cellkjThe corresponding parameter factor. Since the wall thickness variation is smoother than the periodic variation, the gradient constraint of w (r) is no longer required. Finally, selection is made in the MMA solver
Figure BDA0002557908090000077
And
Figure BDA0002557908090000078
an optimized porous structure with both smooth periodicity and wall thickness variation can be obtained. Since the optimization period function p (r) is fixed and the structural porosity monotonically increases with the increase of the wall thickness function w (r), the convergence of the wall thickness optimization is also better achieved. In the experiment, the wall thickness optimization converged to 30 iterations.
The invention discloses a design and optimization system of a porous sheath-shaped heat dissipation structure for 3D printing, and belongs to the field of computer-aided design and industrial design and manufacture. The proposed porous structure is expressed in an implicit function form, and has good connectivity, controllability, mechanical property, thermal property, higher surface area to volume ratio and smoothness. The porous structure is applied to the three-dimensional heat dissipation problem, and an optimized porous structure with continuous geometric change and smooth topological change is obtained. Compared with the existing traditional heat dissipation structure, the porous structure greatly improves the heat dissipation performance, the heat conduction efficiency and the efficiency. The porous structure designed by the invention has the characteristics of smoothness, full connectivity, quasi-self-supporting property and the like, the applicability and manufacturability of the structure are ensured, the porous structure is suitable for a common 3D printing manufacturing method, the internal structure in the printing process does not need to be additionally supported, and the printing time and the printing material can be saved.
Drawings
Fig. 1 is a flow chart of the design and optimization of a three-dimensional porous heat dissipation structure based on three-cycle extremely-small curved surfaces.
Fig. 2 is a diagram of the design and optimization results of a three-cycle minimal curved surface-based three-dimensional porous heat dissipation structure, and a, b, and c are diagrams of the optimization results of three different three-cycle minimal curved surfaces.
Detailed Description
The following further describes a specific embodiment of the present invention with reference to the drawings and technical solutions.
The implementation of the invention can be divided into a plurality of main steps of porous sheath structure function representation, establishment of a heat dissipation problem optimization model and discretization thereof, flow optimization and the like:
method for expressing porous shell-like structure
Firstly, establishing an improved implicit numerical surface:
Figure BDA0002557908090000081
wherein r is a three-dimensional vector, x, y and z are respectively corresponding coordinates of the three-dimensional vector, and P (r) controls the continuous change of the hole period, so that a hole curved surface with smooth transition in space is constructed.
Further, a multi-scale porous sheath-like structure with thickness is constructed: the porous structure with the thickness based on the three-cycle extremely-small curved surface can be obtained by shifting the improved implicit function curved surface to two sides by utilizing a parameter function W (r) for controlling the wall thickness, wherein the two shifted curved surfaces are expressed as follows:
Figure BDA0002557908090000082
Figure BDA0002557908090000083
finally obtaining a porous sheath-like structure based on the three-cycle extremely-small curved surface:
Figure BDA0002557908090000084
to sum up, the value intervals of P (r) (P curved surface are [0.5,2], G curved surface are [0.37,2], D curved surface are [0.5,2], IWP curved surface is [0.48,2]) control the period of the porous structure, w (r) (P curved surface is [0.02,0.95], G curved surface is [0.02,1.35], D curved surface is [0.02,0.7], IWP curved surface is [0.02,2.95]) control the wall thickness of the porous structure.
(II) modeling and optimization based on porous shell-like structure
1. Modeling of heat dissipation problems
The problem of minimizing the weakness of heat dissipation is used here to create a porous structure optimization problem. The inner space of the model is filled by using the constructed porous sheath-like structure with the aim of minimizing the average temperature of the structure and with the constraint of the model volume and boundary conditions, so that the period and the wall thickness of the porous structure have optimal distribution under the condition of the constraint of the material volume.
Based on the above objective, the problem model is established as follows:
Figure BDA0002557908090000091
such that:
Figure BDA0002557908090000092
Figure BDA0002557908090000093
Figure BDA0002557908090000094
where C is the heat dissipation weakness, T is the temperature field, Ω is the given design region, and Φ isThe functional representation of the porous sheath-like structure given above, Q is the heat flux density of the internal heat source, QsIs the Neumann boundaryQThe heat flux in the upper direction of the normal line,
Figure BDA0002557908090000095
is a given temperature on the Directilet boundary, λ is the thermal conductivity;
Figure BDA00025579080900000914
is a vector differential operator which is a function of,
Figure BDA0002557908090000096
x, Y, Z represent unit vectors in the positive direction of the three coordinate axes x, y, and z, respectively;
Figure BDA0002557908090000097
is the corresponding test function and is,
Figure BDA0002557908090000098
Sob1is a first order Sobelov space, V is the volume of the porous structure,
Figure BDA0002557908090000099
is corresponding volume constraint, and adds gradient constraint of periodic variation to avoid the damage of porous structure caused by severe variation of periodic function
Figure BDA00025579080900000910
And formula for calculation of a mode having a gradient
Figure BDA00025579080900000911
H (x) is the Heaviside function, H (x) is 0 when x is negative, otherwise 1, and H (x) is defined as the continuous function H in order to minimize the optimization problem and avoid checkerboard phenomenaη(x) The function is defined as:
Figure BDA00025579080900000912
where, η is the regularization parameter,for controlling the number of non-zero elements in the global stiffness matrix, the parameter η is generally 10-3An interval of intermediate values is defined. Furthermore, the thermal conductivity λ of the material of the porous structure is calculated from the structure function Φ and should be set as:
Figure BDA00025579080900000913
ξ H (phi) is the volume ratio of the solid material, lambdaSAnd λDRepresenting the thermal conductivity of the solid material and the pore portion, respectively.
2. Discretization of optimization problems
In the discretization process, a solving area is subdivided into two sets of uniform grids with different precisions: the temperature field is interpolated with a coarse grid, the model is described with a fine grid and an integral calculation is performed.
Obtaining a discrete form of the optimization problem:
Figure BDA0002557908090000101
such that:
KT＝Q (2.11)
Figure BDA0002557908090000102
Figure BDA0002557908090000103
where T is the temperature field, Q is the heat source and heat flux terms, K is the stiffness matrix, V is the volume of the porous structure,
Figure BDA0002557908090000104
is a corresponding volume constraint, Nb＝nb×nsIs the total number of the thin cells,
Figure BDA0002557908090000105
is the function value of phi of the l-th node in the j-th cell, vbIs the volume of the fine grid cells, G is the global gradient constraint of the structure, | Ω | is ΩVolume, nlIs the number of sub-regions in the design domain,
Figure BDA0002557908090000106
is the ith sub-region omegaiThe number of the thin units in the inner,
Figure BDA0002557908090000107
is the gradient of the periodic function at the ith point in the s-th cell,
Figure BDA0002557908090000108
is the ith sub-region omegaiThe local gradient within the constraint value of the local gradient,
Figure BDA0002557908090000109
is the volume of the ith coarse grid cell, p>0 is a penalty factor for the global gradient constraint and has:
Figure BDA00025579080900001010
the optimization of the period parameter function and the thickness parameter function is converted into the optimization of a limited number of design variables by adopting a global-local RBF interpolation algorithm, and the global-local radial basis interpolation can be simplified into the following form:
Figure BDA00025579080900001011
Figure BDA00025579080900001012
wherein n istIs the total number of control points in the design field omega (value 400), Ni(r) is the corresponding calculable coefficient function,
Figure BDA00025579080900001013
is the value of the periodic function of the control point,
Figure BDA00025579080900001014
is the value of the wall thickness function at the control point. Due to the fact thatThe position of the control points is not changed during the optimization process, so the coefficient function Ni(r) may be calculated prior to optimization.
3. Modeling problem optimization
Only two unknown parameter functions p (r) and w (r) need to be optimized here. The specific optimization process is as follows:
step 1: optimizing the period; firstly, converting function optimization into optimization of interpolation basis function parameters by using a radial basis interpolation method; randomly selecting n in the solution domaintA base point of interpolation
Figure BDA0002557908090000111
There is an interpolation form of (2.14). Thus, the optimization problem is transformed into a parametric variable
Figure BDA0002557908090000112
The optimization problem of (2); finally, the optimization variables are derived by the objective function and the constraint function as follows:
Figure BDA0002557908090000113
Figure BDA0002557908090000114
Figure BDA0002557908090000115
Figure BDA0002557908090000116
wherein the content of the first and second substances,
Figure BDA0002557908090000117
the objective function, the volume constraint and the gradient constraint respectively to the parametric variable PiThe equation for the partial derivative is solved,
Figure BDA0002557908090000118
is an intermediate equation to be calculated in the process of gradient partial derivative calculation; n issIs the number of coarse cells, nbIs the number of thin cells in each thick cell;
Figure BDA0002557908090000119
is phi function value of the l node in the k coarse unit and the j fine unit; from the formula of thermal conductivity
Figure BDA00025579080900001110
Nothing ξkjIs the heat conductivity lambda of the k-th coarse cell and the j-th fine cellkjCorresponding parameter factor ξ, K0Is the initial stiffness matrix. Given the variable sensitivity analysis, the optimized periodic parameter function can be obtained by the well-known MMA method, thus yielding a periodically optimized structure and serving as the initial structure for wall thickness optimization.
Step 2: optimizing the wall thickness; similarly, the control point based on W (r) (variable is
Figure BDA00025579080900001111
) And constructing a wall thickness function W (r) by adopting a radial basis interpolation method, wherein the corresponding sensitivity analysis is as follows:
Figure BDA00025579080900001112
Figure BDA00025579080900001113
wherein the content of the first and second substances,
Figure BDA00025579080900001114
respectively, an objective function, a volume constraint, and a parametric variable WiSolving an equation of partial derivative;
Figure BDA00025579080900001115
is the phi function value of the ith node in the kth coarse cell and the jth fine cell ξkjIs in the k-th coarse cell and the j-th fine cellCoefficient of thermal conductivity lambdakjThe corresponding parameter factor. Substituting into MMA algorithm to obtain the solution of optimization problem.
And the design and optimization of a heat dissipation structure based on the porous structure representation of the three-period extremely-small curved surface are provided. The porous structure is expressed in an implicit function form, and has good connectivity, controllability, higher surface-to-volume ratio, higher smoothness and good mechanical and thermal properties. Various experiments show that the proposed porous structure greatly improves heat dissipation performance, efficiency of heat conduction and efficiency. To achieve high thermal conductivity efficiency, there is a balance between period and wall thickness for a given volume constraint, and the optimized structure period and wall thickness variation is smooth and natural, which facilitates structural stress and fabrication. Compared with the traditional heat dissipation structure and the grid structure, the optimized porous structure has higher heat dissipation efficiency (lower heat dissipation weakness).

Claims (1)

1. A three-cycle extremely-small-curved-surface-based design and optimization method for a three-dimensional porous heat dissipation structure is characterized by comprising the following steps:
method for expressing porous structure
The three-cycle infinitesimal surfaces all have implicit function representations, and the implicit function representation of the P infinitesimal surface is as follows:
Figure FDA0002557908080000011
wherein r is a three-dimensional vector, and x, y and z are corresponding coordinates of the three-dimensional vector respectively;
the periodic parameter function P (r) >0 is directly added into the function representation of the three-period extremely-small curved surface, and in order to ensure that the distance scale of the directional distance field is basically unchanged in the periodic change process, the latent function is improved and represented as follows:
Figure FDA0002557908080000012
wherein P (r) controls the continuous change of the pore period, and a porous curved surface with smooth transition on the space is constructed; other types of three-cycle extremely-small curved surfaces are processed according to the same method;
the three-cycle minimum curved surface-based porous structure with thickness is subjected to the improved implicit function curved surface by controlling the parameter function W (r) of the wall thickness
Figure FDA0002557908080000013
Two offset surfaces are obtained after the offset to two sides, and are expressed as:
Figure FDA0002557908080000014
Figure FDA0002557908080000015
finally, the three-cycle minimal surface-based porous sheath-like structure is represented by an intersection operator continuous function:
Figure FDA0002557908080000016
in the definition, the control of the shape and the periodic holes of the porous structure is realized by introducing a parameter function P (r) >0 for controlling the period and a parameter function W (r) >0 for controlling the wall thickness, and the porous structure with the wall thickness meeting the requirement is finally generated by optimizing the parameter functions P (r) and W (r);
(II) optimization procedure for heat dissipation problem
In the heat dissipation problem under the condition of steady-state heat conduction, after a model heat source and boundary conditions are given, the built porous sheath-like structure is used for filling the internal space of the model, and the optimal distribution of the porous structure period and the wall thickness is solved under the conditions of given material volume constraint and periodic function gradient constraint;
(1) problem modeling
The heat dissipation problem model is established as follows:
Figure FDA0002557908080000021
such that:
Figure FDA0002557908080000022
Figure FDA0002557908080000023
Figure FDA0002557908080000024
where C is the heat dissipation weakness, T is the temperature field, Ω is the given design area, Φ is the functional representation of the porous sheath-like structure given above, Q is the heat flux density of the internal heat source, Q is the heat flux density of the internal heat sourcesIs the Neumann boundaryQThe heat flux in the upper direction of the normal line,
Figure FDA0002557908080000025
is a given temperature on the Directilet boundary, λ is the thermal conductivity;
Figure FDA0002557908080000026
is a vector differential operator which is a function of,
Figure FDA0002557908080000027
x, Y, Z represent unit vectors in the positive direction of the three coordinate axes x, y, and z, respectively;
Figure FDA0002557908080000028
is the corresponding test function and is,
Figure FDA0002557908080000029
Sob1is a first order Sobelov space, V is the volume of the porous structure,
Figure FDA00025579080800000210
is a corresponding volume constraint, in order to avoid the violent change of the periodic function from damaging the porous structure, addsPeriodically varying gradient constraints
Figure FDA00025579080800000211
And formula for calculation of a mode having a gradient
Figure FDA00025579080800000212
H (x) is the Heaviside function, H (x) is 0 when x is negative, otherwise 1, and H (x) is defined as the continuous function H in order to minimize the optimization problem and avoid checkerboard phenomenaη(x) The function is defined as:
Figure FDA00025579080800000213
wherein η is a regularization parameter for controlling the number of non-zero elements in the global stiffness matrix, and the parameter η is 10-3Defining an interval of intermediate values; furthermore, the material thermal conductivity λ of the porous structure is calculated from the structural function Φ, set as:
Figure FDA00025579080800000214
ξ H (phi) is the volume ratio of the solid material, lambdaSAnd λDRespectively representing the thermal conductivity of the solid material and the hole part;
(2) discretization
In the discretization process, a solving area is subdivided into two sets of uniform grids with different precisions: using a coarse grid to interpolate a temperature field, using a fine grid to describe a model and performing integral calculation; the number of coarse cells is nsNumber of fine cells n in each coarse cellbDefault setting is 27; a discrete form of the optimization problem (1.6-1.9) is obtained:
Figure FDA0002557908080000031
such that:
KT＝Q(1.12)
Figure FDA0002557908080000032
Figure FDA0002557908080000033
where T is the temperature field, Q is the heat source and heat flux terms, K is the stiffness matrix, V is the volume of the porous structure,
Figure FDA0002557908080000034
is a corresponding volume constraint, Nb＝nb×nsIs the total number of the thin cells,
Figure FDA0002557908080000035
is the function value of phi of the l-th node in the j-th cell, vbIs the volume of the fine grid cell, G is the global gradient constraint of the structure, | Ω | is the volume of Ω, nlIs the number of sub-regions in the design domain,
Figure FDA0002557908080000036
is the ith sub-region omegaiThe number of the thin units in the inner,
Figure FDA0002557908080000037
is the gradient of the periodic function at the ith point in the s-th cell,
Figure FDA0002557908080000038
is the ith sub-region omegaiThe local gradient within the constraint value of the local gradient,
Figure FDA0002557908080000039
is the volume of the ith coarse grid cell, p>0 is a penalty factor for the global gradient constraint and has:
Figure FDA00025579080800000310
(3) global-local interpolation
The optimization of a periodic parameter function and a thickness parameter function is converted into the optimization of a limited number of design variables by adopting a global-local radial basis interpolation algorithm, and the key idea is to decompose a large coefficient matrix into a weighted small coefficient matrix for solving;
periodic parametric function, first dividing Ω into nlSub-regions
Figure FDA00025579080800000311
And (3) obtaining a local periodic parameter function by radial basis interpolation in a local ellipsoid containing the corresponding sub-region:
Figure FDA00025579080800000312
Figure FDA00025579080800000313
wherein psik(r) is a number fromk(r) defined weight parameter, dk(r)＝‖r-Ck‖2Is interpolated to the ellipsoid central point CkDistance of (1) ()+Is that the truncation function satisfies x>Time 0 (x)+X, otherwise (x)+＝0,Rk(r) is a length function with respect to the radius, Pk(r) is the subregion ΩkA corresponding local periodic parameter function within the local ellipsoid, and defined as:
Figure FDA0002557908080000041
wherein R iski(r)＝(r-Oki)2log(|r-Oki|) is the sheet radial basis function,
Figure FDA0002557908080000042
is the sub-region omegakControl points in the corresponding local ellipsoid, qki(r) is a primary term of coordinates x, y, z, akiAnd bkjAre respectively a quadratic term and aThe coefficient to be solved of the secondary term, m is the number of the primary term, and m is 4;
global-local radial basis interpolation is simplified to the form:
Figure FDA0002557908080000043
wherein n istIs the total number of control points in the design domain omega, and takes 400, Ni(r) is a function of the corresponding coefficients,
Figure FDA0002557908080000044
is the period function value of the control point;
(4) modeling problem optimization
The three-dimensional heat dissipation optimization method based on the constructed optimization problem comprises a period optimization part and a wall thickness optimization part; the period and the wall thickness of the porous structure based on the three-period extremely-small curved surface are independently controlled by a period function P (r) and a wall thickness function W (r), the period optimization is coarse adjustment of the structure, the wall thickness optimization is fine adjustment, and the specific optimization process is as follows:
step 1: optimizing the period; firstly, converting function optimization into optimization of interpolation basis function parameters by using a radial basis interpolation method; randomly selecting n in the solution domaintA base point of interpolation
Figure FDA0002557908080000045
Then there is the interpolation form:
Figure FDA0002557908080000046
conversion of periodic optimization problem into parametric variables
Figure FDA0002557908080000047
The optimization problem of (2); finally, the optimization variables are derived by the objective function and the constraint function as follows:
Figure FDA0002557908080000048
Figure FDA0002557908080000049
Figure FDA0002557908080000051
Figure FDA0002557908080000052
wherein the content of the first and second substances,
Figure FDA0002557908080000053
the objective function, the volume constraint and the gradient constraint respectively to the parametric variable PiThe equation for the partial derivative is solved,
Figure FDA0002557908080000054
is an intermediate equation to be calculated in the process of gradient partial derivative calculation; n issIs the number of coarse cells, nbIs the number of thin cells in each thick cell;
Figure FDA0002557908080000055
is phi function value of the l node in the k coarse unit and the j fine unit; from the formula of thermal conductivity
Figure FDA0002557908080000056
Nothing ξkjIs the heat conductivity lambda of the k-th coarse cell and the j-th fine cellkjCorresponding parameter factor ξ, K0Is an initial stiffness matrix; in MMA solver, by
Figure FDA0002557908080000057
Obtaining an optimized porous structure with a stable period;
step 2: optimizing the wall thickness; similarly, based on the control point of W (r)Measured as
Figure FDA0002557908080000058
And (3) constructing a wall thickness function W (r) by adopting a radial basis interpolation method, and analyzing the corresponding sensitivity as follows:
Figure FDA0002557908080000059
Figure FDA00025579080800000510
wherein the content of the first and second substances,
Figure FDA00025579080800000511
respectively, an objective function, a volume constraint, and a parametric variable WiSolving an equation of partial derivative;
Figure FDA00025579080800000512
is the phi function value of the ith node in the kth coarse cell and the jth fine cell ξkjIs the heat conductivity lambda of the k-th coarse cell and the j-th fine cellkjCorresponding parameter factors; since the wall thickness variation is smoother than the periodic variation, the gradient constraint of w (r) is no longer required; finally, selection is made in the MMA solver
Figure FDA00025579080800000513
And
Figure FDA00025579080800000514
an optimized porous structure is obtained with both smooth periodicity and wall thickness variation.
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