WO2022000132A1 - 基于三周期极小曲面的三维多孔散热结构的设计与优化方法 - Google Patents
基于三周期极小曲面的三维多孔散热结构的设计与优化方法 Download PDFInfo
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- 238000000034 method Methods 0.000 title claims abstract description 46
- 230000017525 heat dissipation Effects 0.000 title claims abstract description 44
- 238000005457 optimization Methods 0.000 claims abstract description 87
- 230000008569 process Effects 0.000 claims abstract description 17
- 239000000463 material Substances 0.000 claims abstract description 9
- 230000000737 periodic effect Effects 0.000 claims description 43
- 238000013461 design Methods 0.000 claims description 25
- 230000008859 change Effects 0.000 claims description 13
- 239000011159 matrix material Substances 0.000 claims description 13
- 230000004907 flux Effects 0.000 claims description 9
- 239000013598 vector Substances 0.000 claims description 9
- 239000011343 solid material Substances 0.000 claims description 6
- 238000004422 calculation algorithm Methods 0.000 claims description 4
- 238000010206 sensitivity analysis Methods 0.000 claims description 4
- 108010074864 Factor XI Proteins 0.000 claims description 3
- 238000009795 derivation Methods 0.000 claims description 3
- 238000009826 distribution Methods 0.000 claims description 3
- 238000012360 testing method Methods 0.000 claims description 3
- 230000007704 transition Effects 0.000 claims description 3
- 239000011148 porous material Substances 0.000 claims description 2
- 230000010354 integration Effects 0.000 claims 1
- 238000004519 manufacturing process Methods 0.000 abstract description 9
- 238000007639 printing Methods 0.000 abstract description 6
- 238000010146 3D printing Methods 0.000 abstract description 3
- 238000011960 computer-aided design Methods 0.000 abstract description 2
- 230000001131 transforming effect Effects 0.000 abstract 1
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- 238000004364 calculation method Methods 0.000 description 2
- 238000010276 construction Methods 0.000 description 1
- 238000001816 cooling Methods 0.000 description 1
- 238000011161 development Methods 0.000 description 1
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- G06F—ELECTRIC DIGITAL DATA PROCESSING
- G06F30/00—Computer-aided design [CAD]
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- G—PHYSICS
- G06—COMPUTING; CALCULATING OR COUNTING
- G06F—ELECTRIC DIGITAL DATA PROCESSING
- G06F30/00—Computer-aided design [CAD]
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- the invention belongs to the field of engineering design and manufacture, and relates to a three-dimensional porous heat dissipation structure design and optimization method, which is suitable for heat dissipation structures of various large-scale construction machinery, radiators such as automobiles, and related components of burning appliances.
- porous structures based on three-period minimal surfaces have been widely used in tissue engineering technology, lightweight manufacturing, and biomedicine.
- Porous structures based on three-period minimal surfaces have the advantages of good connectivity, easy control, high specific strength and stiffness.
- Porous structures based on three-period minimal surfaces are very time-consuming and memory-intensive to represent with polyhedral meshes (tetrahedral or hexahedral), and traditional finite element-based porous structure processing The methods are almost all heuristic and there is no effective optimization. Therefore, there are few researches on the heat dissipation of porous structures based on three-period minimal surfaces.
- an effective representation and optimization method is proposed to obtain the period and wall thickness of the three-period minimal curved porous sheath structure suitable for heat dissipation.
- the main optimization process includes cycle optimization and wall thickness optimization.
- the former is a rough adjustment of the structure, and the latter is a fine adjustment of the structure.
- the porous sheath structure is represented by an implicit function, which is controlled by the periodic parameter function and the wall thickness parameter function; on this basis, the steady-state heat conduction equation with boundary conditions can be conveniently established as a mathematical model by using the function expression.
- the invention proposes an effective expression and optimization method for a porous heat dissipation structure based on a three-period minimal curved surface.
- the porous structure is established by the implicit functional representation of the three-period minimal surface.
- the heat dissipation problem is transformed into a problem of minimizing heat dissipation weakness under given constraints.
- the parametric function is directly solved by the method of global-local interpolation.
- the design and optimization method of the three-dimensional porous heat dissipation structure based on the three-period minimal surface is as follows:
- r is a three-dimensional vector
- x, y, and z are their corresponding coordinates.
- the periodic parameter function P(r)>0 can be directly added to the function representation of the three-period minimal surface.
- the implicit function to be expressed as:
- P(r) controls the continuous change of the hole period, and constructs a porous surface with smooth transition in space; other types of three-period minimal surfaces are processed in the same way.
- the porous structure with thickness based on the three-period minimal surface can control the parameter function W(r) of the wall thickness, and the above improved implicit function surface can be improved. After offset to both sides, the two offset surfaces are expressed as:
- porous sheath-like structure based on the three-period minimal surface is represented by the continuous function of the intersection operator:
- the parameter function P(r)>0 for controlling the period and the parameter function W(r)>0 for controlling the wall thickness are introduced to realize the control of the shape of the porous structure and periodic pores, and by optimizing the parameter function P( r) and W(r) finally generate the desired porous structure with wall thickness.
- the porous structure defined by the above functions inherits the excellent properties of three-period minimal surfaces, such as high surface area to volume ratio, full connectivity, high degree of smoothness, and controllability.
- the high surface-to-volume ratio and full connectivity facilitate heat dissipation of the structure.
- the structure function provides a computable optimization method based on a high degree of controllability of period and wall thickness. Good smoothness and connectivity are beneficial to 3D printing manufacturing, guaranteeing manufacturing accuracy, and can remove excess material (such as excess liquid in SLA) in 3D manufacturing.
- the present invention mainly focuses on the problem of heat dissipation under the condition of steady heat conduction.
- the porous sheath structure constructed above is used to fill the internal space of the model.
- Solve for optimal distributions of porous structure period and wall thickness Solve for optimal distributions of porous structure period and wall thickness.
- C is the heat dissipation weakness
- T is the temperature field
- ⁇ is the given design area
- ⁇ is the functional representation of the porous sheath structure given earlier
- Q is the heat flux density of the internal heat source
- q s is the Neumann boundary ⁇ on Q the heat flux along the normal direction, is the given temperature on the Direchlet boundary
- ⁇ is the thermal conductivity
- X, Y, and Z represent the unit vectors along the positive directions of the three coordinate axes of x, y, and z, respectively
- Sob 1 is the first-order Sobelev space
- V is the volume of the porous structure
- ⁇ is the regularization parameter, which is used to control the number of non-zero elements in the global stiffness matrix.
- volume of the design space discretization process of dual-scale grid, three-dimensional design space design domain first divided into uniform hexahedral Finite Element called coarse cells, the coarse cells for generating a temperature field, a crude unit number n s of decision; then, each coarse cell is further subdivided into smaller hexahedral cells, called fine cells, which are used for more precise geometric calculations such as volume, here, the number of fine cells in each coarse cell n b is set to 27 by default.
- the discrete form of the optimization problem (1.6-1.9) is obtained:
- 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
- v b is the volume of the fine mesh element
- G is the global gradient constraint of the structure
- ⁇ is the volume of ⁇
- n l is the design domain the number of subregions, is the number of fine cells in the ith subregion ⁇ i, is the gradient of the periodic function at the ith point in the sth fine cell
- p>0 is the penalty factor for the global gradient constraint, and has:
- the global-local radial basis interpolation algorithm is used to transform the optimization of the periodic parameter function and the thickness parameter function into the optimization of a limited number of design variables.
- the key idea is to decompose the large coefficient matrix into a weighted small coefficient matrix to solve.
- the local periodic parameter function is obtained by radial basis interpolation in the local ellipsoid (containing the corresponding subregion):
- ⁇ k (r) is the weight parameter defined by ⁇ k (r)
- (*) + is the truncation function
- R k (r) is the length function about the radius
- P k (r) is the local part in the local ellipsoid corresponding to the sub-region ⁇ k period parameter function, and is defined as:
- R ki (r) (rO ki ) 2 log(
- ) is the radial basis function of the thin plate, is the control point in the local ellipsoid corresponding to the sub-region ⁇ k
- q ki (r) is the first-order term of the coordinates x, y, and z
- a ki and b kj are the coefficients to be determined for the quadratic term and the first-order term, respectively
- n t is the total number of control points in the design domain ⁇ (generally 400)
- N i (r) is the corresponding computable coefficient function
- the three-dimensional heat dissipation optimization method proposed based on the optimization problem constructed above includes two parts: cycle optimization and wall thickness optimization.
- the period and wall thickness of the porous structure based on the three-period minimal surface are independently controlled by the periodic function P(r) and the wall thickness function W(r).
- the optimization process is as follows:
- Step 1 Periodic optimization; first, transform the function optimization into the optimization of the parameters of the interpolation basis function by using the radial basis interpolation method; randomly select n t interpolation basis points in the solution domain Then there is an interpolation form:
- n s is the number of coarse cells
- n b is the number of fine cells in each coarse cell
- the thermal conductivity formula is Knowing that ⁇ kj is the parameter factor ⁇ corresponding to the thermal conductivity ⁇ kj in the kth coarse element and the jth fine element
- K 0 is the initial stiffness matrix.
- Step 2 Wall thickness optimization; similarly, the control point based on W(r) (variable is The radial basis interpolation method is used to construct the wall thickness function W(r), and the corresponding sensitivity analysis is as follows:
- the 3D printing-oriented porous sheath heat dissipation structure design and optimization system of the invention belongs to the fields of computer-aided design and industrial design and manufacture.
- the proposed porous structure is expressed in the form of an implicit function and exhibits good connectivity, controllability, mechanical properties, thermal properties, high surface area-to-volume ratio, and smoothness. Applying the proposed porous structure to the 3D heat dissipation problem results in an optimized porous structure with continuous geometric changes and smooth topological changes. Compared with the existing traditional heat dissipation structure, the porous structure greatly improves the heat dissipation performance and the efficiency and effectiveness of heat conduction.
- the porous structure designed by the invention has the characteristics of smoothness, full connectivity, quasi-self-supporting, etc., which ensures the applicability and manufacturability of this type of structure.
- This type of porous structure is suitable for common 3D printing manufacturing methods.
- the printing process The internal structure does not require additional support, which can save printing time and printing materials.
- Figure 1 is a flow chart of the design and optimization of a three-dimensional porous heat dissipation structure based on a three-period minimal surface.
- Figure 2 shows the results of the design and optimization of the three-dimensional porous heat dissipation structure based on the three-period minimal surface
- a, b, and c are the optimization results of three different three-period minimal surfaces.
- the implementation of the present invention can be divided into several main steps such as porous sheath structure function representation, establishment of an optimization model of heat dissipation problem and its discretization, and optimization process:
- r is a three-dimensional vector, and x, y, and z are their corresponding coordinates, respectively.
- P(r) controls the continuous change of the hole period, and constructs a hole surface with a smooth transition in space.
- a multi-scale porous sheath-like structure with thickness is constructed: the porous structure with thickness based on the three-period minimal surface can be shifted to the two sides by using the parameter function W(r) that controls the wall thickness. Obtained, the two offset surfaces are represented as:
- the value range of P(r) (P surface is [0.5, 2], the value range of G surface is [0.37, 2], the value range of D surface is [0.5, 2], and the value range of IWP surface is [0.5, 2].
- the value interval of W(r) (the value interval of P surface is [0.02, 0.95], the value interval of G surface is [0.02, 1.35], and the value interval of D surface is [0.02, 1.35])
- the value range of ⁇ is [0.02, 0.7]
- the value range of IWP surface is [0.02, 2.95]) to control the wall thickness of the porous structure.
- the problem of minimizing heat dissipation weakness is used to establish the optimization problem of porous structure. That is, with the goal of minimizing the average temperature of the structure and the constraints of the model volume and boundary conditions, the porous sheath structure constructed above is used to fill the internal space of the model, so that the period and wall thickness of the porous structure have a given material volume constraint. optimal distribution.
- C is the heat dissipation weakness
- T is the temperature field
- ⁇ is the given design area
- ⁇ is the functional representation of the porous sheath structure given earlier
- Q is the heat flux density of the internal heat source
- q s is the Neumann boundary ⁇ on Q the heat flux along the normal direction, is the given temperature on the Direchlet boundary
- ⁇ is the thermal conductivity
- X, Y, and Z represent the unit vectors along the positive directions of the three coordinate axes of x, y, and z, respectively
- Sob 1 is the first-order Sobelev space
- V is the volume of the porous structure
- ⁇ is the regularization parameter, which is used to control the number of non-zero elements in the global stiffness matrix.
- the solution area is subdivided into two sets of uniform grids with different precisions: coarse grids are used to interpolate the temperature field, and fine grids are used to describe the model and perform integral calculations.
- 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
- v b is the volume of the fine mesh element
- G is the global gradient constraint of the structure
- ⁇ is the volume of ⁇
- n l is the design domain the number of subregions, is the number of fine cells in the ith subregion ⁇ i, is the gradient of the periodic function at the ith point in the sth fine cell
- p>0 is the penalty factor for the global gradient constraint, and has:
- the global-local radial basis interpolation can be simplified to the following form:
- n t is the total number of control points in the design domain ⁇ (value 400)
- N i (r) is the corresponding computable coefficient function
- Step 1 Periodic optimization; first, transform the function optimization into the optimization of the parameters of the interpolation basis function by using the radial basis interpolation method; randomly select n t interpolation basis points in the solution domain Then there is the interpolation form of (2.14). In this way, the periodic optimization problem is transformed into a parametric variable The optimization problem of ; finally, through the derivation of the objective function and the constraint function with respect to the optimization variables, as follows:
- n s is the number of coarse cells
- n b is the number of fine cells in each coarse cell
- the thermal conductivity formula is Knowing that ⁇ kj is the parameter factor ⁇ corresponding to the thermal conductivity ⁇ kj in the kth coarse element and the jth fine element
- K 0 is the initial stiffness matrix.
- Step 2 Wall thickness optimization; similarly, the control point based on W(r) (variable is ), the radial basis interpolation method is used to construct the wall thickness function W(r), and the corresponding sensitivity analysis is as follows:
- the design and optimization of the heat dissipation structure based on the porous structure representation of the three-period minimal surface is proposed.
- the porous structure is expressed in the form of an implicit function, and has good connectivity, controllability, high surface-to-volume ratio, high smoothness, and good mechanical and thermal properties.
- Various experiments show that the proposed porous structure greatly improves the heat dissipation performance, the efficiency and effectiveness of heat conduction.
- the optimized structural period and wall thickness changes are smooth and natural, which is beneficial to structural stress and fabrication .
- the optimized porous structure has higher heat dissipation efficiency (lower heat dissipation weakness).
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Abstract
基于三周期极小曲面的三维多孔散热结构的设计与优化方法,属于计算机辅助设计领域。首先,通过三周期极小曲面的隐式函数表示来建立多孔结构。其次,根据稳态热传导方程,将散热问题转化为在给定约束条件下散热弱度的最小化问题。然后,利用全局‐局部插值的方法直接求解参数函数。最后,对建模问题进行周期优化和壁厚优化,得到了具有光滑周期和壁厚变化的优化后的多孔鞘状结构。本发明的多孔结构大大改善了散热性能、热传导的效率和效能。本发明设计的多孔结构具有光滑性、全连通性、可控性、准自支撑性等特性,这些特质确保了该类结构的适用性和可制造性,适用于3D打印制造方法,打印过程的内部结构无需额外支撑,节省打印时间和打印材料。
Description
本发明属于工程设计与制造领域,涉及一种三维多孔散热结构设计与优化方法,适用于各种大型工程机械的散热结构,汽车等散热器,以及燃具相关部件。
如何构造轻质、高效的散热结构在各个工程领域受到了广泛的关注。传统的散热器结构不能达到较高的导热效率。通过对多孔结构的研究,可以有效地提高结构的导热性能,然而,表示和优化方法成为制约其进一步发展的技术瓶颈。
传统的散热结构设计,依靠热学的基本理论和实际经验,难以解决复杂结构的散热问题。随后,出现了利用树状拓扑结构、桁架/框架结构、微结构等多孔结构的拓扑优化方法来处理上述问题,这些多孔结构可用于计算冷却通道的高自由度拓扑结构。然而,这些方法有一个共同的问题,它们需要大量的设计变量,并且由于耗时的重新网格化,优化的代价是昂贵的。
近年来,基于三周期极小曲面的多孔结构在组织工程技术、轻量化制造、生物医学等领域得到了广泛的应用。基于三周期极小曲面的多孔结构具有良好的连通性、易控制、高比强度和刚度等优点。基于三周期极小曲面的多孔结构(特别是对于大型复杂的多孔结构)用多面体网格(四面体或六面体)来表示是非常耗时和耗费内存的,并且传统的基于有限元的多孔结构处理方法几乎都是启发式的,没有有效的优化,因此,基于三周期极小曲面的多孔结构在散热方面的研究较少。
基于上述目的,提出了一种有效的表示和优化方法,来获得适合于散热的三周期极小曲面多孔鞘状结构的周期和壁厚。主要优化过程包括周期优化和壁厚优化。前者是对结构的粗调整,后者是对结构的细调整。首先,用隐函数表示多孔鞘状结构,隐函数由周期参数函数和壁厚参数函数控制;在此基础上,利用函数表达式可以方便地将具有边界条件的稳态热传导方程直接建立为数学模型,然后将模型的优化问题转化为求解上述两个连续参数函数;最后从离散化 的角度出发,利用隐函数表示和径向基插值可以有效地计算出这两个参数函数,而不需要重新网格化,最终得到了具有光滑周期和壁厚变化的优化的散热多孔结构。
发明内容
本发明提出了一种基于三周期极小曲面的多孔散热结构的有效表达和优化方法。首先,通过三周期极小曲面的隐式函数表示来建立多孔结构。其次,根据稳态热传导方程,将散热问题转化为在给定约束条件下散热弱度的最小化问题。然后,利用全局-局部插值的方法直接求解参数函数。最后,我们对建模问题进行周期优化和壁厚优化,得到了具有光滑周期和壁厚变化的优化后的多孔鞘状结构。
本发明采用的技术方案是:
基于三周期极小曲面的三维多孔散热结构的设计与优化方法,方法如下:
(一)多孔结构表示方法
常用的三周期极小曲面大都具有隐函数表示,以P极小曲面为例:
其中,r为三维向量,x,y,z分别为其对应坐标。
周期参数函数P(r)>0可以直接加入到三周期极小曲面的函数表示中,为了使周期变化过程中有向距离场的距离尺度基本不变,我们改进了隐函数表示为:
其中,P(r)控制了孔洞周期的连续变化,构造在空间上具有光滑过渡的多孔曲面;其它类型的三周期极小面按照相同的方法进行处理。
最终由交算子连续函数表示基于三周期极小曲面的多孔鞘状结构:
在上述定义中,引入控制周期的参数函数P(r)>0和控制壁厚的参数函数W(r)>0来实现对多孔结构的形状和周期孔洞的控制,并且通过优化参数函数P(r)和W(r)最终生成满足需求的有壁厚的多孔结构。
上述函数定义的多孔结构继承了三周期极小曲面的优良特性,如高表面积体积比、全连通性、高度光滑和可控性。高的表面体积比和全连通性有利于结构的散热。结构函数基于周期和壁厚的高度可控性提供了可计算的优化方法。良好的光滑性和连通性有利于3D打印制造,保证制造的准确性,并且可以在3D制造中去除多余的材料(如SLA中的多余液体)。
(二)散热问题优化过程
本发明主要关注稳态热传导情况下的散热问题,给定模型热源及边界条件后,利用上述构建的多孔鞘状结构来填充模型内部空间,在给定材料体积约束和周期函数梯度约束情况下,求解多孔结构周期和壁厚的优化分布。
1.问题模型建立
基于上述目的,散热问题模型建立如下:
使得:
其中,C是散热弱度,T是温度场,Ω是给定设计区域,Φ是前面给出的多孔鞘状结构的函数表示,Q是内热源的热流密度,q
s是Neumann边界Γ
Q上沿法线方向的热通量,
是Direchlet边界上的给定温度,λ是导热系数;
是向量微分算子,
X、Y、Z分别表示沿着x、y、z三个坐标轴正方向的单位向量;
是对应的测试函数,
Sob
1 是一阶索勃列夫空间,V是多孔结构的体积,
是对应的体积约束,为了避免周期函数的剧烈变化破坏多孔结构,加入了周期变化的梯度约束
并且有梯度的模的计算公式
H(x)是Heaviside函数,当x为负时,H(x)=0,否则为1,为了使优化问题可微并且避免棋盘格现象,将H(x)定义为连续函数H
η(x),函数定义为:
其中η是正则化参数,用于控制全局刚度矩阵中非零元素的数量,一般取参数η=10
-3定义中间值的区间。此外,多孔结构的材料导热系数λ是由结构函数Φ计算得到的,应该设置为:
ξ=H(Φ)是固体材料的体积比,λ
S和λ
D分别表示固体材料和孔洞部分的导热系数。
2.离散化
离散化过程中采用双尺度网格,设计域的三维设计空间首先被划分为均匀六面体有限元,称为粗单元,粗单元用于生成温度场,粗单元的数量n
s的由设计空间的体积决定;然后,每个粗单元被进一步细分为更小的六面体单元,称为细单元,细单元用于进行体积等更精确的几何计算,这里,每个粗单元中的细单元的数量n
b默认设置为27。得到优化问题(1.6-1.9)的离散形式:
使得:
KT=Q(1.12)
其中,T是温度场,Q是热源和热通量项,K是刚度矩阵,V是多孔结构的体积,
是对应的体积约束,N
b=n
b×n
s是细单元的总数,
是第j个细单元内的第l个结点的Φ函数值,v
b是细网格单元的体积,G是结构的全局梯度约束,‖Ω‖是Ω的体积,n
l是设计域中子区域的数量,
是第i个子区域Ω
i内的细单元的个数,
是周期函数在第s个细单元内第i点的梯度,
是第i个子区域Ω
i内的局部梯度约束值,
是第i个粗网格单元的体积,p>0是全局梯度约束的惩罚因子,并且有:
3.全局-局部插值
采用全局-局部的径向基插值算法,将周期参数函数和厚度参数函数的优化转化为有限数量的设计变量的优化,其关键思想是将大型系数矩阵分解成加权的小型系数矩阵求解。
其中,ψ
k(r)是由ω
k(r)定义的权重参数,d
k(r)=‖r-C
k‖
2是插值点到椭球体中心点C
k的距离,(*)
+是截断函数满足x>0时(x)
+=x,否则(x)
+=0,R
k(r)是关于半径的长度函数,P
k(r)是子区域Ω
k对应的局部椭球体内的局部周期参数函数,并定义为:
其中,R
ki(r)=(r-O
ki)
2log(|r-O
ki|)是薄板径向基函数,
是子区域Ω
k对应的局部椭球体内的控制点,q
ki(r)是坐标x、y、z的一次项,a
ki和b
kj分别是二次项和一次项的待求系数,m是一次项的个数(默认为m=4)。
全局-局部径向基插值可以被简化为如下形式:
4.建模问题优化
基于上述构建的优化问题提出的三维散热优化方法包括周期优化和壁厚优化两部分。基于三周期极小曲面的多孔结构的周期和壁厚分别由周期函数P(r)和壁厚函数W(r)独立控制,周期优化是对结构的粗调整,壁厚优化是细调整,具体优化流程如下:
其中,
分别是目标函数、体积约束和梯度约束对参数变量P
i求偏导数的等式,
是梯度求偏导数过程中要计算的中间等式;n
s是粗单元的数量,n
b是每个粗单元内细单元的数量;
是第k个粗单元,第j个细单元内的第l个结点的Φ函数值;由导热系数公式
知ξ
kj是是第k个粗单元,第j个细单元内的导热系数λ
kj对应的参数因子ξ;K
0是初始刚度矩阵。在MMA求解 器中,通过
就可以获得周期平稳变化的优化多孔结构。由于壁厚函数W(r)是固定的,结构孔隙率整体上随周期函数P(r)的增大而增大,因此易于实现周期优化的收敛性。在我们的实验中,周期优化收敛于70次迭代。
其中,
分别是目标函数、体积约束对参数变量W
i求偏导数的等式;
是第k个粗单元,第j个细单元内的第l个结点的Φ函数值;ξ
kj是是第k个粗单元,第j个细单元内的导热系数λ
kj对应的参数因子。由于壁厚变化比周期变化更平稳,因此不再需要W(r)的梯度约束。最后,在MMA求解器中选择
和
就可以获得同时具有光滑周期和壁厚变化的优化多孔结构。由于优化周期函数P(r)是固定的,结构孔隙率随壁厚函数W(r)的增大而单调增大,因此壁厚优化的收敛性也较好实现。实验中,壁厚优化收敛于30次迭代。
本发明的面向3D打印的多孔鞘状散热结构设计与优化系统,属于计算机辅助设计、工业设计制造领域。所提出的多孔结构以隐函数形式表示,具有良好的连通性、可控性、力学性能、热学性能、较高的表面积体积比和光滑性。将所提出的多孔结构应用于三维散热问题,得到了一个具有连续几何变化和平滑拓扑变化的优化多孔结构。与现有的传统散热结构相比,该多孔结构大大改善了散热性能、热传导的效率和效能。该发明设计的多孔结构具有的光滑性、全连通性、准自支撑性等特性,确保了该类结构的适用性和可制造性,该类多孔结构适用于常用的3D打印制造方法,打印过程的内部结构无需额外支撑,可以节省打印时间和打印材料。
图1是基于三周期极小曲面的三维多孔散热结构的设计与优化流程图。
图2是基于三周期极小曲面的三维多孔散热结构的设计与优化结果图,a,b,c是三种不同三周期极小曲面优化结果图。
以下结合附图和技术方案,进一步说明本发明的具体实施方式。
本发明实施具体可分为多孔鞘状结构函数表示,建立散热问题优化模型及其离散化,优化流程等几个主要步骤:
(一)多孔壳状结构表示方法
首先建立改进的隐含数曲面:
其中,r为三维向量,x,y,z分别为其对应坐标,P(r)控制了孔洞周期的连续变化,构造在空间上具有光滑过渡的孔洞曲面。
进而,构造具有厚度的多尺度多孔鞘状结构:基于三周期极小曲面的有厚度的多孔结构可以通过利用控制壁厚的参数函数W(r)对上述改进隐函数曲面向两侧偏移后得到,两个偏移曲面表示为:
最终得到基于三周期极小曲面的多孔鞘状结构:
综上可知,P(r)(P曲面的取值区间为[0.5,2],G曲面的取值区间为[0.37,2],D曲面的取值区间为[0.5,2],IWP曲面的取值区间为[0.48,2])控制多孔结构的周期,W(r)(P曲面的取值区间为[0.02,0.95],G曲面的取值区间为[0.02,1.35],D曲面的取值区间为[0.02,0.7],IWP曲面的取值区间为[0.02,2.95])控制多孔结构的壁厚。
(二)基于多孔壳状结构的建模及优化
1.散热问题建模
这里使用最小化散热弱度问题来建立多孔结构优化问题。即以结构平均温度最小化为目标,以模型体积、边界条件为约束,利用上述构建的多孔鞘状结构来填充模型内部空间,使得给定材料体积约束情况下,多孔结构的周期和壁厚具有最优化的分布。
基于上述目的,问题模型建立如下:
使得:
其中,C是散热弱度,T是温度场,Ω是给定设计区域,Φ是前面给出的多孔鞘状结构的函数表示,Q是内热源的热流密度,q
s是Neumann边界Γ
Q上沿法线方向的热通量,
是Direchlet边界上的给定温度,λ是导热系数;
是向量微分算子,
X、Y、Z分别表示沿着x、y、z三个坐标轴正方向的单位向量;
是对应的测试函数,
Sob
1是一阶索勃列夫空间,V是多孔结构的体积,
是对应的体积约束,为了避免周期函数的剧烈变化破坏多孔结构,加入了周期变化的梯度约束
并且有梯度的模的计算公式
H(x)是Heaviside函数,当x为负时,H(x)=0,否则为1,为了使优化问题可微并且避免棋盘格现象,将H(x)定义为连续函数H
η(x),函数定义为:
其中,η是正则化参数,用于控制全局刚度矩阵中非零元素的数量,一般取参数η=10
-3定义中间值的区间。此外,多孔结构的材料导热系数λ是由结构函数Φ计算得到的,应该设置为:
ξ=H(Φ)是固体材料的体积比,λ
S和λ
D分别表示固体材料和孔洞部分的导热系数。
2.优化问题的离散化
离散化过程中将求解区域细分成两套精度不同的均匀网格:用粗网格去插值温度场,用细网格去描述模型和进行积分计算。
得到优化问题的离散形式:
使得:
KT=Q(2.11)
其中,T是温度场,Q是热源和热通量项,K是刚度矩阵,V是多孔结构的体积,
是对应的体积约束,N
b=n
b×n
s是细单元的总数,
是第j个细单元内的第l个结点的Φ函数值,v
b是细网格单元的体积,G是结构的全局梯度约束,‖Ω‖是Ω的体积,n
l是设计域中子区域的数量,
是第i个子区域Ω
i内的细单元的个数,
是周期函数在第s个细单元内第i点的梯度,
是第i个子区域Ω
i内的局部梯度约束值,
是第i个粗网格单元的体积,p>0是全局梯度约束的惩罚因子,并且有:
采用全局-局部的RBF插值算法,将周期参数函数和厚度参数函数的优化转化为有限数量的设计变量的优化,全局-局部径向基插值可以被简化为如下形式:
其中,n
t是设计域Ω内控制点的总数(取值400),N
i(r)是相应的可计算的系数函数,
是控制点的周期函数值,
是控制点的壁厚函数值。由于在优化过程中不改变控制点的位置,所以系数函数N
i(r)可以在优化前提前计算。
3.建模问题优化
这里只需对两个未知参数函数P(r)和W(r)进行优化。具体优化流程如下:
步骤1:周期优化;首先,利用径向基插值的方法将函数优化转化为插值基函数参数的优化;在求解域中随机选择n
t个插值基点
则有(2.14)的插值形式。如此,周期优化问题就转换为了对参数变量
的优化问题;最后,通过对目标函数和约束函数关于优化变量进行求导,如下:
其中,
分别是目标函数、体积约束和梯度约束对参数变量P
i求偏导数的等式,
是梯度求偏导数过程中要计算的中间等式;n
s是粗单元的数量,n
b是每个粗单元内细单元的数量;
是第k个粗单元,第j个细单元内的第l个结点的Φ函数值;由导热系数公式
知ξ
kj是是第k个粗单元,第j个细单元内的导热系数λ
kj对应的参数因子ξ;K
0是初始刚度矩阵。给定变量的敏感度分析,优化的周期参数函数可以通过众所周知的MMA方法获得,这样就得到了周期优化后的结构,并且作为壁厚优化的初始结构。
其中,
分别是目标函数、体积约束对参数变量W
i求偏导数的等式;
是第k个粗单元,第j个细单元内的第l个结点的Φ函数值;ξ
kj是是第k个粗单元, 第j个细单元内的导热系数λ
kj对应的参数因子。代入MMA算法最终得到优化问题的解。
提出基于三周期极小曲面的多孔结构表示的散热结构设计和优化。多孔结构以隐函数形式表示,具有良好的连通性、可控性、较高的表面体积比、较高的光滑性和良好的力学、热学性能。各种实验表明,所提出的多孔结构大大改善了散热性能、热传导的效率和效能。为了获得较高的导热效率,在给定的体积约束下,周期与壁厚之间存在平衡,且优化后的结构周期和壁厚的变化是平滑的、自然的,这有利于结构应力和制造。优化后的多孔结构与传统的散热结构以及格子结构相比,具有更高的散热效率(较低的散热弱度)。
Claims (1)
- 一种基于三周期极小曲面的三维多孔散热结构的设计与优化方法,其特征在于,步骤如下:(一)多孔结构表示方法三周期极小曲面都具有隐函数表示,P极小曲面的隐函数表示如下:其中,r为三维向量,x,y,z分别为其对应坐标;周期参数函数P(r)>0直接加入到三周期极小曲面的函数表示中,为了使周期变化过程中有向距离场的距离尺度基本不变,改进隐函数,表示为:其中,P(r)控制了孔洞周期的连续变化,构造在空间上具有光滑过渡的多孔曲面;其它类型的三周期极小曲面按照相同的方法进行处理;最终由交算子连续函数表示基于三周期极小曲面的多孔鞘状结构:在上述定义中,引入控制周期的参数函数P(r)>0和控制壁厚的参数函数W(r)>0来实现对多孔结构的形状和周期孔洞的控制,并且通过优化参数函数P(r)和W(r)最终生成满足需求的有壁厚的多孔结构;(二)散热问题优化过程稳态热传导情况下的散热问题,给定模型热源及边界条件后,利用上述构建的多孔鞘状结构来填充模型内部空间,在给定材料体积约束和周期函数梯度约束情况下,求解多孔结构周期和壁厚的优化分布;(1)问题模型建立散热问题模型建立如下:使得:其中,C是散热弱度,T是温度场,Ω是给定设计区域,Φ是前面给出的多孔鞘状结构的函数表示,Q是内热源的热流密度,q S是Neumann边界Γ Q上沿法线方向的热通量, 是Direchlet边界上的给定温度,λ是导热系数; 是向量微分算子, X、Y、Z分别表示沿着x、y、z三个坐标轴正方向的单位向量; 是对应的测试函数, Sob 1是一阶索勃列夫空间,V是多孔结构的体积, 是对应的体积约束,为了避免周期函数的剧烈变化破坏多孔结构,加入了周期变化的梯度约束 并且有梯度的模的计算公式 H(x)是Heaviside函数,当x为负时,H(x)=0,否则为1,为了使优化问题可微并且避免棋盘格现象,将H(x)定义为连续函数H η(x),函数定义为:其中η是正则化参数,用于控制全局刚度矩阵中非零元素的数量,取参数η=10 -3定义中间值的区间;此外,多孔结构的材料导热系数λ是由结构函数Φ计算得到的,设置为: ξ=H(Φ)是固体材料的体积比,λ S和λ D分别表示固体材料和孔洞部分的导热系数;(2)离散化离散化过程中将求解区域细分成两套精度不同的均匀网格:用粗网格去插 值温度场,用细网格去描述模型和进行积分计算;粗单元的数量为n s,每个粗单元中的细单元的数量n b默认设置为27;得到优化问题(1.6-1.9)的离散形式:使得:KT=Q(1.12)其中,T是温度场,Q是热源和热通量项,K是刚度矩阵,V是多孔结构的体积, 是对应的体积约束,N b=n b×n s是细单元的总数, 是第j个细单元内的第l个结点的Φ函数值,v b是细网格单元的体积,G是结构的全局梯度约束,‖Ω‖是Ω的体积,n l是设计域中子区域的数量, 是第i个子区域Ω i内的细单元的个数, 是周期函数在第s个细单元内第i点的梯度, 是第i个子区域Ω i内的局部梯度约束值, 是第i个粗网格单元的体积,p>0是全局梯度约束的惩罚因子,并且有:(3)全局-局部插值采用全局-局部的径向基插值算法,将周期参数函数和厚度参数函数的优化转化为有限数量的设计变量的优化,其关键思想是将大型系数矩阵分解成加权的小型系数矩阵求解;其中,ψ k(r)是由ω k(r)定义的权重参数,d k(r)=‖r-C k‖ 2是插值点到椭球体 中心点C k的距离,(*) +是截断函数满足x>0时(x) +=x,否则(x) +=0,R k(r)是关于半径的长度函数,P k(r)是子区域Ω k对应的局部椭球体内的局部周期参数函数,并定义为:其中,R ki(r)=(r-O ki) 2log(|r-O ki|)是薄板径向基函数, 是子区域Ω k对应的局部椭球体内的控制点,q ki(r)是坐标x、y、z的一次项,a ki和b kj分别是二次项和一次项的待求系数,m是一次项的个数,m=4;全局-局部径向基插值被简化为如下形式:(4)建模问题优化基于上述构建的优化问题提出的三维散热优化方法包括周期优化和壁厚优化两部分;基于三周期极小曲面的多孔结构的周期和壁厚分别由周期函数P(r)和壁厚函数W(r)独立控制,周期优化是对结构的粗调整,壁厚优化是细调整,具体优化流程如下:其中, 分别是目标函数、体积约束和梯度约束对参数变量P i求偏导数的等式, 是梯度求偏导数过程中要计算的中间等式;n s是粗单元的数量,n b是每个粗单元内细单元的数量; 是第k个粗单元,第j个细单元内的第l个结点的Φ函数值;由导热系数公式 知ξ kj是是第k个粗单元,第j个细单元内的导热系数λ kj对应的参数因子ξ;K 0是初始刚度矩阵;在MMA求解器中,通过 获得周期平稳变化的优化多孔结构;
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CN116013443A (zh) * | 2023-03-22 | 2023-04-25 | 中国空气动力研究与发展中心计算空气动力研究所 | 一种传热特性预测方法、装置、设备及可读存储介质 |
CN116187107A (zh) * | 2023-04-27 | 2023-05-30 | 中南大学 | 三维地温场动态数值模拟方法、设备及介质 |
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CN117332523A (zh) * | 2023-09-27 | 2024-01-02 | 之江实验室 | 一种基于非局域时空模型的机器人结构件优化方法及装置 |
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