A surface flow fiber bundle topology optimization method

Abstract

The present application relates to a kind of surface flowing fiber cluster topology optimization method;The present application uses material interpolation method to determine surface flowing model, implicit two-dimensional manifold of surface flowing is defined on bottom flow form, porous medium is filled on implicit two-dimensional manifold, while artificial Darcy friction coefficient is introduced to surface Navier-Stokes equation, and surface flowing fiber cluster topology optimization is realized by filling porous medium to fixed two-dimensional manifold.

Description

Technical Field

[0001] This invention relates to the field of fluid mechanics, and in particular to a method for optimizing fiber bundle topology in surface flow. Background Technology

[0002] In computational design related to fluid structures, surface flow can significantly reduce computational costs. Flow within channels attached to the solid walls of a device can be described as surface flow on curved surfaces corresponding to the external shapes of industrial equipment and microfluidic devices. The flow surface corresponding to the external shape of a device with a fully slip boundary can be described as surface flow separated from the volume flow. The fully slip boundary can be approximated and realized through chemical coatings or physical structural solid surfaces to obtain extreme hydrophobicity, manipulate the boundary velocity of the flow using optimal control methods, and generate gas layers between solid and liquid phases based on the Leidenfrost phenomenon.

[0003] Topology optimization is currently considered one of the most reliable methods for determining material distribution in structures that meet given structural performance criteria. Regarding the question of whether a topology optimization method can be implemented that matches the surface flow model and the implicit two-dimensional manifold of the defined pattern, if such topology optimization can be achieved, the design space and design degrees of freedom of the flow problem will be further expanded, incorporating the design domain of surface flow into the design space, where the design domain is an implicit two-dimensional manifold. Therefore, this paper proposes a fiber bundle topology optimization method for surface flow. Summary of the Invention

[0004] The main technical problem solved by this invention is to provide a fiber bundle topology optimization method for surface flow. It uses material interpolation to determine the surface flow model, defines an implicit two-dimensional manifold for surface flow on the bottom manifold, fills the implicit two-dimensional manifold with a porous medium, and introduces an artificial Darcy friction coefficient into the surface Navier-Stokes equation. By filling the fixed two-dimensional manifold with a porous medium, the fiber bundle topology optimization of surface flow is achieved.

[0005] To solve the above-mentioned technical problems, one technical solution adopted by the present invention is: to provide a fiber bundle topology optimization method for surface flow, wherein, it includes:

[0006] The surface flow model is determined by material interpolation. An implicit two-dimensional manifold for surface flow is defined on the bottom manifold. A porous medium is filled on the implicit two-dimensional manifold, and an artificial Darcy friction coefficient is introduced into the surface Navier-Stokes equations.

[0007] The artificial Darcy friction coefficient is obtained based on the constitutive relation of the porous media model and is assumed to be proportional to the fluid velocity.

[0008]

[0009] Where α is the reverse osmosis coefficient, Γ represents the implicit two-dimensional manifold, and x Γ Let Γ represent a node on an implicit two-dimensional manifold. When the porosity of the porous model is 0, it represents a solid material with infinite permeability. Due to the infinite friction, its fluid velocity is 0. When the porosity is infinite, it corresponds to a hollow structure used to control liquid transport. The reverse osmosis coefficient can be expressed as follows:

[0010]

[0011] Where, γ p ∈{0,1} is a binary distribution defined on Γ, where 0 and 1 represent solid and liquid, respectively. D It is the design domain for surface flow, Γ s It is the material density γ p For a solid with a value of 0, Γ F It is the material density γ p When the liquid is 1, Γ D , Γ S , Γ F Satisfy Γ D ∪Γ S ∪Γ F =Γ and Γ represents the design domain; the binary variable γ in the design domain p Allowed to vary continuously between [0, 1], the relaxed binary variable is called the material density of the porous medium. According to the above formula describing the porous medium, the material interpolation of the porous medium can be described by a convex q-parameterized form:

[0012]

[0013] Where, α s and α f α and β represent the reverse osmosis coefficients for solids and liquids, respectively; q is used to adjust the convexity of the material interpolation; for liquids, the reverse osmosis coefficient is 0, and α... f =0, for solids, α s Theoretically, it is infinite, so we choose a value much greater than the liquid density ρ and assign it to α. s To ensure sufficient accuracy in approximating solid materials and the stability of numerical calculations, based on numerical experiments, the optimized parameters in material interpolation are set as follows: q = 1, α = 1. s Take 10 4 α s Satisfy α s >>ρ.

[0014] As an improvement of the present invention, the design variables in surface flow topology optimization include design variables of implicit two-dimensional manifold, design variables of surface flow model, and coupling of design variables.

[0015] As a further improvement of the present invention, the design variables of the implicit two-dimensional manifold are defined as continuously varying design variables [0, 1] on the base manifold. The design variables of the implicit two-dimensional manifold are used to describe the normal displacement distribution of the implicit two-dimensional manifold relative to the base manifold. The surface flow model is defined in a variable design domain. The result of fiber bundle topology optimization is regarded as a second-order structure containing the implicit two-dimensional manifold and the surface flow model. To ensure the smoothness of the two-dimensional manifold and the stability of the solution, a surface partial differential equation filter is applied to the design variables of the implicit two-dimensional manifold.

[0016]

[0017] Where, d m d is the design variable of the two-dimensional manifold. f r is the filter design variable. m It is the filter radius and is a constant, ∑ is the base manifold used to define the implicit two-dimensional manifold, x ∑ Represents a point on ∑. and div ∑ These are the gradient and divergence operators defined on ∑, n τ∑ =n ∑ *τ ∑ It is a tangent perpendicular to Σ. The outward unit normal vector, n ∑ and τ ∑ Representing ∑ normal and Tangential unit vector, A d d is the amplitude of the displacement between the implicit two-dimensional manifold and the fixed two-dimensional manifold, and it is non-negative. m d is a value between [0, 1]. f Is The values ​​between;

[0018] After the filtering operation, the filtered implicit two-dimensional manifold can be described as follows:

[0019]

[0020] Where Γ is an implicit two-dimensional manifold, x Γ Represents a point on Γ; from the above equation, through mapping It can be determined to be differentially identical topological. and Also of the same topology, according to the above equation, the Jacobian matrix of the implicit two-dimensional manifold in the base manifold curvilinear coordinates can be transformed into:

[0021]

[0022] in, Represent its determinant;

[0023] The variational form of the surface partial differential equation filter is defined in a first-order Sobolev space on the surface ∑, which is obtained according to the Galerkin method:

[0024] Find such that,

[0025]

[0026] in, It is d f trial function, Let be a first-order Sobolev space defined on ∑. Let represent a second-order Lebesque space defined on ∑.

[0027] As a further improvement of the present invention, the design variable of the surface flow model is represented by the material density defined on the invisible two-dimensional manifold. The material density is obtained by filtering the surface differential equation and thresholding the material density defined on the two-dimensional manifold. The design value of the surface flow model is a continuous value between [0, 1]. The thresholding is used to remove grayscale regions in the model. The surface flow partial differential equation filter of the design value is obtained by solving the following surface partial differential equation:

[0028]

[0029] Where Γ is the design variable, Γ f It is the filtered design variable, r f It is the filter radius and is a constant; and div Γ These are the tangential gradient and the three-dimensional operator defined on the implicit two-dimensional manifold, respectively, n. τΓ =n Γ *τ Γ It is a tangent perpendicular to f. The outward unit normal vector, n Γ and τ Γ Representing the Γ normal and The threshold projection of the tangential unit vector and the filter design variable is expressed as:

[0030]

[0031] Where β and ξ are the parameters of the threshold projection, which are obtained through numerical calculation and testing;

[0032] The variational form of the surface partial differential equation filter is defined in a first-order Sobolev space on the surface Γ, obtained according to the Galerkin method:

[0033] Find such that,

[0034] in, It is γ f trial function, Let be a first-order Sobolev space defined on ∑. Let denote a second-order Lebesque space defined on Γ.

[0035] As a further improvement of the present invention, the coupling of the design variables is defined on the implicit two-dimensional manifold by the design variables of the surface flow model, and the coupling relationship is achieved by transforming the tangential gradient operator. Tangential divergence operator div Γ and unit normal vector n Γ This is obtained from the form defined on the bottom flow type Σ;

[0036] Tangential gradient operator The conversion is based on the following formula:

[0037]

[0038] Among them, P Γ It is the projection defined on the tangential space of Γ0, with the normal unit vector n. Γ It can be represented in the following form:

[0039]

[0040] Where ||·||² is the second-order modulus of the vector, and the design variable d is used for filtering. f The superscript distinguishes the transformed unit normal vector from its original form. This identification method is then applied to other transformed operators and variables, where the projection P... Γ It can then be expressed in the following form:

[0041]

[0042] Where I is the second-order unit tensor, and the superscript T indicates the transpose of the tensor or vector;

[0043] Tangential gradient operator The conversion is based on the following formula:

[0044]

[0045] Based on the transformation method of the tangential gradient algorithm, the tangential divergence operator div Γ Convert to

[0046]

[0047] Where tr is the rank of the tensor;

[0048] Because of the tangential gradient operator Depends on d f d f The first variational form of is as follows, for

[0049] For d f div operator Γ The first-order form is as follows, for

[0050]

[0051] Because of d f If the differential is topologically identical, then the Riemannian metric is introduced, which satisfies the differential on the base manifold and the differential on the implicit two-dimensional manifold;

[0052]

[0053] in, and Let Γ and ∑ be the differentials of the boundary curves, respectively;

[0054] Based on the transformation form of the tangential gradient operator and Same topology, tangential gradient operator The transformation is based on the variational form of the filter derived from the partial differential equation of the surface, which yields the coupling of the design variables.

[0055] Find, such that,

[0056]

[0057] gradient operator on Γ Transformed into form.

[0058] As a further improvement of the present invention, the surface Navier-Stokes equations on implicit two-dimensional manifolds are as follows: Newton's surface fluid motion equations describe two-dimensional manifolds with a codimensional of 1 in Euclidean space. Based on the laws of conservation of momentum and mass, they evolve into surface Navier-Stokes equations to describe incompressible surface flow:

[0059]

[0060] Where u is the fluid velocity, p is the fluid pressure, ρ is the fluid density, and η is the shear viscosity coefficient of the fluid on the implicit two-dimensional manifold, u·n Γ=0 is the tangential constraint on fluid velocity. The tangential constraint is passively applied because the fluid flows in space on the two-dimensional manifold Γ, and the fluid velocity is a vector in the tangential space of Γ.

[0061] Solving the Navier-Stokes equations for the surface is possible because the fluid velocity and pressure at the interface, boundary, and points on the two-dimensional manifold Γ are known.

[0062]

[0063] in, The velocity distribution is known and depends on the boundary or interface on the surface ∑. The known flow velocity l at the location v，∑ l v，Γ It is the boundary on the Γ plane, satisfying At the same time, satisfy l v，Γ It is the interface on the Γ surface, l s，Γ It is a boundary curve with open boundary conditions, and satisfies p 0，Γ The fluid pressure is known and depends on a finite set of points on the surface ∑. The specific fluid pressure p 0，∑ ; A finite set of points; It is a finite set of points;

[0064] In the above equation, when the known flow velocity is 0, the boundary conditions at the inlet or interface of the flow velocity degenerate into no-slip boundary conditions:

[0065]

[0066] Among them, Down, It is the non-slip part of the curved surface;

[0067] The variational form of the surface Navier-Stokes equations does not consider the function space without tangential constraints on fluid velocity. These tangential constraints are obtained using the Lagrange multiplier method. According to the Galerkin method, the variational form of the surface Navier-Stokes equations is:

[0068] Find such that,

[0069]

[0070] Where λ is a Lagrange multiplier that imposes a tangential constraint on the velocity; and It is a trial function of u, p, and λ; dΓ and dl ΓThese are the differential and boundary curve of the two-dimensional manifold, respectively. The Lagrange multipliers in the above equation are used to provide constraints on the tangential velocity of the fluid, which is equivalent to uniformly distributing a force field in the normal direction of the two-dimensional manifold. This force field cancels out the centrifugal force, Coriolis force, and Euler force from the fluid particles in the normal direction of the two-dimensional manifold to satisfy the tangential constraint.

[0071] Because l v，Γ With l v，∑ They are topologically identical. and They are topologically identical, l v，∑ and It is certain. and p 0，Γ respectively with p 0，∑ Since they are topologically identical, and based on the coupling relationship of the design variables, the variational form of the above equation can be transformed into a form defined on the two-dimensional manifold ∑:

[0072] Find such that,

[0073]

[0074] As a further improvement of the present invention, the design objective for surface flow fiber bundle topology optimization is expressed in the following universal form:

[0075]

[0076] Where A and B are the integrands of the design objective J, and based on the coupling relationship of the design variables, the design objective in the above equation can be transformed into a definition on the base manifold ∑:

[0077]

[0078] In the above formula, the unit tangential vector τ Γ exist The relationship above:

[0079]

[0080] The design goal then becomes:

[0081]

[0082] Based on the transformed design objectives, an adjoint analysis of fiber bundle topology optimization was performed on the functional space defined on the bottom manifold.

[0083] As a further improvement of the present invention, the bottom manifold, together with the implicit two-dimensional manifold and the surface flow model, constitutes a fiber bundle, where ∑ is the bottom manifold and Γ×[0,1] are the fibers in the fiber bundle.

[0084] proj1 is the natural projection:

[0085] proj1: ∑×(Γ×[0,1])→∑satisfying proj1(X) ∑ , (X Γ , Γ p ))=proj1(X ∑ , (d f (X ∑ ), γ p ))=X ∑ ,for φ1 is a model with the same topology, φ1: ∑→Γ×[0,1] satisfying φ1(X ∑ )=(X ∑ γ p )=(d f (X ∑ ), γ p For a model with the same topology φ2: Γ×[0,1]→∑×(Γ×[0,1], satisfying φ2(X Γ γ p )=(X ∑ (X ∑ γ p ))=(X ∑ , (d f (X ∑ ), γ p )),for

[0086] Based on the above representation, the fiber bundle topology optimization for surface flow is constructed as follows:

[0087] Find for(∑×(Γ×[0,1]),∑,proj1,Γ×[0,1]),

[0088] Maximize or minimize with Under the following constraints:

[0089]

[0090] Where J0 is the initial value of the design variable J; area constraints and volume constraints are applied to the surface flow model and the implicit two-dimensional manifold, respectively; s is the area ratio of the surface flow model; v is the volume ratio of the regions enclosed by the implicit two-dimensional manifold and the bottom flow model; and S0∈(0,1) and v0∈(0,1) are specific area and volume ratios, respectively.

[0091] As a further improvement of the present invention, the adjoint analysis for fiber bundle topology optimization is solved using an iterative procedure based on gradient information, wherein the adjoint sensitivity is used to determine the relevant gradient information, which is obtained by adjoint analysis of the design objective, area constraint and volume constraint.

[0092] Based on the continuous adjoint analysis method, the evolution process of the sensitivity J of the adjoint design target is as follows:

[0093]

[0094] Where, γ fa With d fa These are the filter design variables γ f With d f The adjoint variable is δ, which is a first-order variational operator. The adjoint variable is derived from the adjoint equation in the variational formula.

[0095] The variational formula for the adjoint equation of the Naver-Stokes equations on surfaces evolves as follows:

[0096] Find such that

[0097]

[0098] Among them, u a p a , λ a These are the accompanying variables of u, p, and λ, respectively; They are u a p a , λ a The trial function;

[0099] γ and d m It is derived from the adjoint equation of surface partial differential filtering, and its variational formula is as follows:

[0100] Find γ fa It belongs to a first-order Sobolev space defined on ∑, such that

[0101]

[0102] Find (d fa It belongs to a first-order Sobolev space defined on ∑, such that

[0103]

[0104] in, and They are γfa With d fa The trial function;

[0105] For area constraints, the evolution of the associated sensitivity s is as follows:

[0106]

[0107] Based on the above equation, the accompanying sensitivity δ(s|Γ|) changes from s|Γ|=∫ Γ γ p The adjoint analysis of dΓ evolved as follows:

[0108]

[0109] In the above equation, by solving for γ and d m The variational formula of the adjoint equation of the surface partial differential filter can be used to obtain the adjoint variable γ. fa With d m :

[0110] Find (γ fa It belongs to a first-order Sobolev space defined on ∑, such that

[0111]

[0112] Find (d fa It belongs to a first-order Sobolev space defined on ∑, such that

[0113]

[0114] Based on the evolution of the accompanying sensitivity s, the accompanying sensitivity δ|Γ| changes from |Γ|=∫ Γ The association analysis of 1dΓ evolved from this;

[0115] In the following equation, by solving for γ and d m The variational formula of the adjoint equation of the surface partial differential filter can be used to obtain the adjoint variable γ. fa With d m :

[0116]

[0117] Find (γ fa It belongs to a first-order Sobolev space defined on ∑, such that

[0118]

[0119] Find (d fa It belongs to a first-order Sobolev space defined on ∑, such that

[0120]

[0121] For volume constraints, the evolution of the associated sensitivity v is as follows:

[0122]

[0123] In the above formula, by solving d m The variational formula of the adjoint equation of the surface partial differential filter yields the adjoint variable d. f Accompany variable d fa The variational formula for solving the adjoint equation of surface partial differential filtering is d. m Evolved from:

[0124]

[0125] in, (d fa It belongs to a first-order Sobolev space defined on ∑.

[0126] By evolving the design variables using adjoint sensitivity, we obtain γ and d. m The process involves iterative evolution to determine the fiber bundle structure.

[0127] The beneficial effects of this invention are as follows: Compared with the prior art, this invention uses material interpolation to determine the model of surface flow, defines an implicit two-dimensional manifold for surface flow on the bottom manifold, fills the implicit two-dimensional manifold with porous media, and introduces an artificial Darcy friction coefficient into the surface Navier-Stokes equation. By filling the fixed two-dimensional manifold with porous media, the fiber bundle topology optimization of surface flow is achieved. Attached Figure Description

[0128] Figure 1 This is a schematic diagram of the fiber bundles for surface flow in an embodiment of this application;

[0129] Figure 2 This is a schematic diagram of an implicit two-dimensional manifold surface partial differential equation filter defined on the bottom manifold ∑ according to an embodiment of this application;

[0130] Figure 3 This is a schematic diagram illustrating the filtering and projection process of the surface model design values ​​in an embodiment of this application;

[0131] Figure 4 For the embodiments of this application in The unit tangent vector τ on Γ The unit normal vector n on ∑ ∑and tangential gradient A diagram illustrating the relationships between them;

[0132] Figure 5 A sketch of the Taylor-Hood mesh for an embodiment of this application, showing the linear and quadratic elements of the base manifold ∑ based on the discretization of quadrilateral elements and the mapped mesh on the implicit two-dimensional manifold Γ.

[0133] Figure 6 This is a bottom manifold sketch of fiber bundle topology optimization for curved channels in an embodiment of this application, where ∑ D It is the design domain, ∑ F It is a channel domain;

[0134] Figure 7 For the different embodiments of this application A d The fiber bundle topology optimization of the curved channel of the value includes the distribution of filtered design variables, the curved flow model projected onto the bottom manifold, and the implicit two-dimensional manifold with surface manifold composed of the bottom manifold and the pattern fiber bundle, where the arrows indicate the distribution of fluid velocity.

[0135] Figure 8 For the embodiments of this application, the amplitude A is... d The convergence value of the fiber bundle design objective is obtained by setting the elements sequentially to {0, 1, 2, 3, 4, 5, 6, 7};

[0136] Figure 9 The convergence history of the curved channel results in the embodiments of this application includes the fiber bundle evolution process;

[0137] Figure 10 For the fiber bundle topology optimization of the curved channel at different velocities U0 at the entrance of the embodiments of this application, including the filtered design variable d f The distribution of the curved channel γ projected onto ∑ p The pattern and the fiber bundle consisting of the pattern and the two-dimensional manifold, where the arrows indicate the distribution of fluid velocity;

[0138] Figure 11 Corresponding to the embodiments of this application Figure 10 The design target values ​​for the fiber bundles are shown in bold, where optimized entries are shown in bold.

[0139] Figure 12 This is a bottom manifold sketch of fiber bundle topology optimization for four-terminal elements according to an embodiment of this application, where ∑ D It is the design domain, ∑ F It is a channel domain;

[0140] Figure 13 For different inlet velocity amplitudes U0 in the embodiments of this application, the fiber bundle topology of the four-terminal element is optimized, where the arrows indicate the fluid velocity distribution;

[0141] Figure 14 To optimize the fiber bundle topology of the bottom manifold by transforming the surface manifold from a square to a sphere while maintaining area conservation in this embodiment of the application;

[0142] Figure 15 Examples of this application Figure 12 The diagram shows the fluid pressure distribution on the surface of the fiber bundle.

[0143] Figure 16 In this embodiment of the application, the surface manifold of the bottom manifold is transformed from a cylinder to a Möbius ring for fiber bundle topology optimization while maintaining area conservation.

[0144] Figure 17 Examples of this application Figure 16 The diagram shows the fluid pressure distribution on the surface of the fiber bundle. Detailed Implementation

[0145] The topology of surface flow can be described as Figure 1 The fiber bundle shown is a concept in differential geometry; the topology of surface flow can be described as... Figure 1 The fiber bundle shown is an implicit two-dimensional manifold used to define the surface flow model, where ∑ is the bottom manifold and Γ is the implicit two-dimensional manifold used to define the surface flow model. This is a surface flow model, where u is the fluid velocity of the surface flow. n is the known fluid velocity at the Γ boundary. Γ τ is the unit normal vector of Γ. Γ yes The unit tangential vector at n τΓ =n Γ ×τ Γ yes The unit vector outside, x ∑ Let x represent a point on ∑. Γ Let Γ represent a point.

[0146] This application's embodiments primarily demonstrate a fiber bundle topology optimization method for laminar surface flows with low and medium Reynolds numbers. The surface flow can be turbulent with high Reynolds numbers. The fiber bundle consists of a bottom manifold and fibers defined on it, where the manifold represents a topological space that is locally isotopic in Euclidean space. For surface flows, the manifold and its domain correspond to the fibers in the fiber bundle. If there exists a two-dimensional manifold that is isomorphic to the fibers, it can be set as the bottom manifold of the fiber bundle.

[0147] In this embodiment of the application, to ensure the smoothness of the two-dimensional manifold and the stability of the solution, such as... Figure 2 The surface partial differential equation filter shown is applied to the design variables of an invisible two-dimensional manifold:

[0148]

[0149] Where, d m It is the design variable of the two-dimensional manifold; d f It is the filter design variable; r m It is the filter radius, which is a constant; ∑ is the base manifold used to define the implicit two-dimensional manifold; x ∑ Represents a point on ∑; and div ∑ These are the gradient and divergence operators defined on ∑; n τ∑ =n ∑ *τ ∑ It is a tangent perpendicular to ∑ The outward unit normal vector, n ∑ , τ ∑ Representing ∑ normal and Tangential unit vector; A d d is the amplitude of the displacement between the implicit two-dimensional manifold and the fixed two-dimensional manifold; it is non-negative. m d is a value between [0, 1]. f Is The values ​​between.

[0150] In this embodiment, the design variables of the surface flow model are represented by the material density defined on the invisible two-dimensional manifold, such as... Figure 3 As shown, the material density is obtained by filtering the surface differential equation and thresholding the projection on the material density defined on the two-dimensional manifold.

[0151] The design values ​​of the surface flow model are also continuous values ​​between [0, 1]. Threshold projection is used to remove grayscale regions from the obtained pattern. The surface flow partial differential equation filter of the design values ​​can be obtained by solving the following surface partial differential equation:

[0152]

[0153] Where Γ is the design variable; Γ f These are the filtered design variables; r f It is the filter radius, which is a constant. and div Γ These are the tangential gradient and the three-dimensional operator, defined on the implicit two-dimensional manifold, respectively; n τΓ =n Γ *τ Γ It is a sectional plane perpendicular to Γ. The outward unit normal vector, n Γ , τ Γ Representing the Γ normal and Tangential unit vector; the threshold projection of the filter design variables is expressed as:

[0154]

[0155] Here, β and ξ are the parameters of the threshold projection, which are obtained through numerical calculation and testing.

[0156] The variational form of the surface partial differential equation filter considers a first-order Sobolev space defined on the surface Γ, and it can be obtained using the Galerkin method:

[0157] Find such that

[0158]

[0159] in, It is γ f trial function, Let be a first-order Sobolev space defined on ∑; Let denote a second-order Lebesque space defined on Γ.

[0160] The basic form of the design objective in this embodiment is as follows: The design objective of the surface flow fiber bundle topology optimization problem can be expressed in the following universal form:

[0161]

[0162] Where A and B are the integrands of the design objective J, and based on the coupling relationship of the design variables, the design objective in the above equation can be transformed into a definition on the base manifold ∑:

[0163]

[0164] In the above formula, the unit tangential vector τ Γ exist The above satisfies Figure 4 The relationship outlined in the text:

[0165]

[0166] The design goal can be transformed into:

[0167]

[0168] Based on the transformed design objectives, the adjoint analysis of the fiber bundle topology optimization problem can be performed in the functional space defined on the bottom manifold.

[0169] The fiber bundle topology optimization problem in this embodiment is as follows: a fiber bundle is formed by a bottom manifold, an implicit two-dimensional manifold, and a surface flow model, where ∑ is the bottom manifold and Γ×[0,1] are the fibers in the fiber bundle. This fiber bundle is as follows: Figure 5 As shown, proj1 is the natural projection: proj1: ∑×(Γ×[0,1])→∑satisfying proj1(X ∑ , (X Γ , Γ p ))=proj1(X ∑ , (d f (X ∑ ), γ p ))=X ∑ ,for φ1 is a model with the same topology, φ1: ∑→Γ×[0,1] satisfying φ1(X ∑ )=(X ∑ γ p )=(d f (X ∑ ), γ p For a model with the same topology φ2: Γ×[0,1]→∑×(Γ×[0,1], satisfying φ2(X Γ γ p )=(X ∑ (X ∑ γ p ))=(X ∑ , (d f (X ∑ ), γ p )),for

[0170] Based on the above introduction, the fiber bundle topology optimization for surface flow can be constructed in the following form:

[0171] Find for(∑×(Γ×[0,1]),∑,proj1,Γ×[0,1])

[0172] Maximize or minimize with

[0173] Under the following constraints:

[0174]

[0175] Where J0 is the initial value of the design variable J; to adjust this optimization problem, area constraints and volume constraints are added to the surface flow model and the implicit two-dimensional manifold, respectively; s is the area ratio of the surface flow model; v is the volume ratio of the regions enclosed by the implicit two-dimensional manifold and the bottom flow model; S0∈(0,1) and v0∈(0,1) are specific area and volume ratios, respectively.

[0176] The numerical example of iterative solution of fiber bundle topology optimization in this invention can be solved by an iterative process described by pseudocode to solve the fiber bundle topology optimization problem of surface flow, which includes a loop for iterative solution; the surface finite element method can be used to solve the variational formulas of the relevant partial differential equations and adjoint equations.

[0177] When solving surface flow problems on implicit two-dimensional manifolds filled with porous media using the surface finite element method, the Lagrange multiplier method can be used to apply tangential constraints to the fluid velocity. To avoid numerical singularities caused by null denominators, the second modulus of the vector function is approximated in the numerical implementation as ||f||² → (f 2 +ε0) 1 / 2 , where f is a vector function and ε0 is a floating-point precision value.

[0178] To ensure the well-posedness of the numerical solutions of the variational formulas for the surface Navier-Stokes equations and their adjoint equations, a conditionally satisfied Taylor-Hood method was used. Linear elements were used to interpolate the design variables of the surface flow mode and solve the variational formulas and corresponding adjoint equations of the surface partial differential filter for these design variables. Quadratic elements were used to interpolate the design variables of the implicit two-dimensional manifold and solve the variational formulas and corresponding adjoint equations of the surface partial differential filter for these design variables. The Taylor-Hood method mesh and the linear and quadratic elements based on the quadrilateral element-based bottom flow discretization have been used in [the study / implementation]. Figure 5 The mapping mesh is drawn on the implicit two-dimensional manifold.

[0179] During the iteration process, the projection parameter β, which is initially set to 1, is doubled every 30 iterations. When the maximum number of iterations is reached or the design objective changes on average over 5 consecutive iterations, and the residuals of the area and volume constraints are simultaneously satisfied, the loop stops, and the design variables are updated using the moving asymptote method.

[0180] Within this invention, numerical examples are used to demonstrate the proposed fiber bundle topology optimization method. Specifically, fiber bundle topology optimization is performed on surface flows defined on several different bottom manifolds, including curved channels and four-end element planes, curved surfaces deformed from squares to spheres, and curved surfaces deformed from cylinders to Möbius rings. That is, the numerical examples used to demonstrate the proposed fiber bundle topology optimization method include curved channel examples, four-end element plane examples, and deformable interface flow examples.

[0181] The design goal is set as a combination of viscous power dissipation and pressure drop between the inlet and outlet:

[0182]

[0183] Where ω is the weight of viscous dissipation with a value of 9 / 10, pressure drop has a weight of 1 / 10, fluid density and dynamic viscosity are specified as single variables, surface flow is driven by the boundary velocity at the inlet, and the arc length function is parabolic with a magnitude of U0. The exit is set to open boundary conditions, while the other boundaries are set to no-slip boundary conditions.

[0184] In this embodiment, the curved channel calculation example is for a curved channel within the design domain ∑ D and fluid domain Σ F Perform fiber bundle topology optimization on the plane ∑, such as Figure 6 As shown; for different amplitudes A d Optimized fiber bundles and their components are obtained, such as Figure 7 As shown, the distribution includes the velocity vector, where the area and volume fractions are set to s0 = 0.3 and v0 = 0, respectively. Figure 7 Listed Figure 8 The target value of the optimization result obtained from; from Figure 7 From this, we can conclude that a higher A d The value helps reduce viscous dissipation and pressure drop in flexural flow because the design space for the fiber bundle topology optimization problem of surface flow can be amplified by increasing the value of the amplitude parameter, especially when A... d When the value is 0, the fiber bundle topology optimization problem degenerates into a topology optimization problem of curved channels on a plane, such as... Figure 8 As shown in (a). By setting the volume fraction to 0, the implicit two-dimensional manifold Γ is derived with the absolute values ​​of the positive and negative parts of the closed volumes on both sides of Σ being equal, which can be seen from... Figure 7 The filter design variable d shown f This was confirmed in the distribution (a1, b1, c1, d1, e1, f1, g1, and h1). Figure 8 In the diagram, (a1) to (a3) ​​are A d =0 results; (b1)~(b3) are A d =1; (c1)~(c3) are A d =2; (d1)~(d3) are A d =3; (e1)~(e3) are A d =4; (f1)~(f3) are A d =5; (g1)~(g3) are A d =6; (h1)~(h3) are A d The result is 7. For Figure 8 c in A d The convergence history of the result = 2, the objective value, and the area and volume constraints has been plotted in [the relevant database]. Figure 9The image includes a picture of the fiber bundle evolution process. The convergence history shows that the numerical solution for the fiber bundle topology optimization problem of the curved channel converges stably. However, the convergence history exhibits jumps in the objective value and area constraint, caused by the projection parameter β in the iterative equations. Conversely, the convergence history of the volume constraint is smooth without jumps because no projection operation was applied to regularize the design variables of the implicit 2D manifold. This can be achieved by selecting the amplitude parameter A. d =2, further investigation was conducted on the fiber bundle topology optimization problem at the entrance of the curved channel with different velocity amplitudes. By setting U0 sequentially to {1×100, 2×102, 5×102, 8×102}, the optimization results are as follows. Figure 10 As shown, (a1) to (a3) ​​are U0 = 1 × 100 0 The results (b1) to (b3) are U0 = 2 × 10 2 The results (c1) to (c3) are U0 = 5 × 10 2 The results (d1) to (d3) are U0 = 8 × 10 2 As a result, since a larger U0 value corresponds to a stronger Reynolds effect in surface flow, increasing the velocity at the inlet can enhance convection in surface flow. Therefore, different fiber bundles are derived, such as... Figure 10 As shown. To confirm the optimality of the fiber bundles produced by the bent flow, by... Figure 11 The target value and Figure 10 Compare the results in the middle; from Figure 11 Starting with the lowest value of the design target in each row (marked in bold), the optimized performance of the derived fiber bundle can be confirmed.

[0185] The four-terminal element planar computation example in this application specifically addresses the fiber bundle topology optimization problem with a planar manifold as the base, by adjusting the amplitude A... d The four-terminal components were further studied by setting them to 0 and 2 respectively, and the computational domain was set as follows: Figure 12 The design domain ∑ is shown D and fluid domain ∑ F The plane formed by setting its area and volume fractions to s0 = 0.4 and v0 = 0, and the optimization results obtained for different velocities at the inlet are as follows: Figure 13 As shown.

[0186] In the deformable interface flow example of this application embodiment, when a square is deformed into a hemisphere, the surface manifold mode branches into two branches; implicit two-dimensional manifold contraction is used to correct the channels corresponding to the surface flow mode, such as... Figure 14 As shown, its corresponding bottom manifold model is as follows: Figure 14 As shown in (a1), 14(b1), and 14(c1); the distribution of the implicit two-dimensional manifold filter design variables is as follows: Figure 14As shown in (a2), 14(b2), and 14(c2); the projection model on the bottom manifold is as follows: Figure 14 As shown in (a3), 14(b3), and 14(c3); the evolution of fiber bundles is as follows Figure 14 As shown in (a4), 14(b4), and 14(c4), when the hemisphere further deforms into a sphere, the two branches merge, causing a portion of the no-slip boundary to be removed, and the surface flow evolves into a closed mode with two vortices. The mechanism by which the fiber bundles evolve with the deformation of the bottom manifold is that the fluid tends to move in the shortest path, the channels widen, and the removal of the no-slip boundary helps reduce viscous dissipation and pressure drop. Furthermore, as the bottom manifold deforms from a square to a sphere, the viscous loss and pressure drop of the surface flow decrease with the reduction of the optimized fiber bundle pattern, which can be seen from... Figure 15 The correspondence shown Figure 14 The fluid pressure distribution of the fiber bundle surface flow in (a4), 14(b4), and 14(c4) is confirmed.

[0187] Further proof demonstrates that fiber bundle topology optimization is achieved on the bottom manifold by first deforming the cylinder into a strip, and then into a Möbius ring, as shown below. Figure 16 As shown, Figure 16 (a1), 16(b1), 16(c1), 16(d1), and 16(e1) are schematic diagrams of the bottom manifold; Figure 16 (a2), 16(b2), 16(c2), 16(d2), 16(e2) are implicit two-dimensional manifolds for the distribution of the filter design variables; Figure 16 (a3), 16(b3), 16(c3), 16(d3), 16(e3) are the sets of projection modes on the base manifold; Figure 16 (a4), 16(b4), 16(c4), 16(d4), and 16(e4) are evolutionary models on fiber bundles, such as... Figure 16 As shown in (a1~e1), the bars are marked with dimensions to minimize viscous dissipation and pressure drop in surface flow. The area, volume fraction, and size parameters remain unchanged. The optimized surface manifold and implicit two-dimensional manifold of the fiber bundle are shown below. Figure 16 As shown in (a2~a4), 16(b2~b4), 16(c2~c4), 16(d2~d4), and 16(e2~e4), the distribution of velocity vectors is represented by arrows. In particular, the implicit two-dimensional manifold evolved in the Möbius ring is broken by simultaneously setting the entrance to an exit with the same known velocity. The purpose of this setting is to make the derived implicit two-dimensional manifold orientable and to remove the singularity of the normal direction on the non-orientable Möbius ring. Figure 17 Corresponding to Figure 16The fluid pressure distributions on the surface of the fiber bundles shown in (a4), 16(b4), 16(c4), 16(d4), and 16(e4) are illustrated. The derived model and implicit 2D manifold defined on the cylinder consist of a circular channel and an asymmetric surface. The cylinder is then opened, successively evolving into a semi-cylindrical shape, a strip shape, and a Möbius ring, until it is closed again into a Möbius ring, maintaining area conservation. The derived model and implicit 2D manifold are defined on the bottom manifold, corresponding to the sequential evolution of the open manifold. The cylinder is entirely composed of a grooved model and an asymmetric surface. This asymmetry helps the derived implicit 2D manifold satisfy volume constraints when the volume fraction is 0, which is caused by the asymmetry of surface flow convection. Asymmetry is beneficial for shortening the surface flow path and reducing viscous losses and pressure drop.

[0188] The above description is merely an embodiment of the present invention and does not limit the patent scope of the present invention. Any equivalent structural or procedural transformations made based on the content of the present invention specification and drawings, or direct or indirect applications in other related technical fields, are similarly included within the patent protection scope of the present invention.

Claims

1. A method for optimizing the topology of fiber bundles for surface flow, characterized in that, include: The surface flow model is determined by material interpolation. An implicit two-dimensional manifold for surface flow is defined on the bottom manifold. A porous medium is filled on the implicit two-dimensional manifold. At the same time, an artificial Darcy friction coefficient is introduced into the surface Navier-Stokes equations. The artificial Darcy friction coefficient is obtained based on the constitutive relation of the porous media model and is assumed to be proportional to the fluid velocity. Where α is the impermeability coefficient, f represents the implicit two-dimensional manifold, and x Γ Let Γ represent a node on an implicit two-dimensional manifold. When the porosity of the porous model is 0, it represents an infinitely permeable solid material, where the fluid velocity is 0 due to infinite friction. When the porosity is infinite, it corresponds to a hollow structure, allowing for zero impermeability of the liquid transported. Impermeability is represented as follows: Where, γ p ∈{0,1} is a binary distribution defined on Γ, where 0 and 1 represent solid and liquid, respectively. D It is the design domain for surface flow, Γ s It is the material density γ p For a solid with a value of 0, Γ F It is the material density γ p When the liquid is 1, Γ D , Γ S , Γ F Satisfy Γ D ∪Γ S UΓ F =Γ and Γ represents the design domain, where no region is both solid and liquid. S ∩Γ F , and Γ=Γ D ; Binary variable γ in the design domain p Allowed to vary continuously between [0, 1], the relaxed binary variable is called the material density of the impermeable material. According to the description of impermeability above, the interpolation of the impermeable material is described by convexity and q parameterization as follows: Where, α s and α f Representing the impermeability of solids and liquids respectively; q is the convexity used to adjust the interpolation; for liquids, the impermeability is 0, and α... f =0, for solids, α s Theoretically, it is infinite, so we choose a value much greater than the liquid density ρ and assign it to α. s To ensure the stability of numerical calculations with sufficient precision, numerical tests show that q is 1 and α... s Take 10 4 ρ satisfies α s >>ρ.

2. The fiber bundle topology optimization method for surface flow according to claim 1, characterized in that, Design variables in topology optimization of surface flow include design variables of implicit two-dimensional manifolds, design variables of surface flow models, and coupling of design variables.

3. The fiber bundle topology optimization method for surface flow according to claim 2, characterized in that, The design variables of the implicit two-dimensional manifold are defined as continuously varying design variables [0, 1] on the base manifold. These design variables describe the normal displacement distribution of the implicit two-dimensional manifold relative to the base manifold. The surface flow model is defined in a variable design domain. The result of fiber bundle topology optimization is considered as a second-order structure containing both the implicit two-dimensional manifold and the surface flow model. To ensure the smoothness of the two-dimensional manifold and the stability of the solution, a surface partial differential equation filter is applied to the design variables of the implicit two-dimensional manifold. Where, d m d is the design variable of the two-dimensional manifold. f r is the filter design variable. m It is the filter radius and is a constant, ∑ is the base manifold used to define the implicit two-dimensional manifold, x ∑ Denotes a point on ∑. and div ∑ These are the gradient and divergence operators defined on ∑, n τ∑ =n ∑ *τ ∑ It is a tangent perpendicular to ∑ The outward unit normal vector, n ∑ and τ ∑ Representing ∑ normal and Tangential unit vector, A d d is the amplitude of the displacement between the implicit two-dimensional manifold and the fixed two-dimensional manifold, and it is non-negative. m d is a value between [0, 1]. f Is The values ​​between; After the filtering operation, the filtered implicit two-dimensional manifold is described as follows: Where Γ is an implicit two-dimensional manifold, x Γ Represents a point on Γ; from the above equation, through mapping d f :∑7→Γ,x Γ =d f n Σ +x ∑ , It can be determined to be differentially identical topological. and Also of the same topology, according to the above equation, the Jacobian matrix on the implicit two-dimensional manifold in the base manifold curvilinear coordinates can be transformed into: in, Represent its determinant; The variational form of the surface partial differential equation filter is defined in a first-order Sobolev space on the surface ∑, which is obtained according to the Galerkin method: in, It is d f trial function, Let be a first-order Sobolev space defined on ∑. Let represent a second-order Lebesque space defined on ∑.

4. The fiber bundle topology optimization method for surface flow according to claim 3, characterized in that, The design variable of the surface flow model is represented by the material density defined on an invisible two-dimensional manifold. The material density is obtained by filtering the surface differential equation and thresholding the material density defined on the two-dimensional manifold. The design values ​​of the surface flow model are continuous values ​​between [0, 1]. The thresholding is used to remove grayscale regions from the model. The surface flow partial differential equation filtering of the design variable is obtained by solving the following surface partial differential equation: Where γ is the design variable, γ f It is the filtered design variable, r f It is the filter radius and is a constant; and div Γ These are the tangential gradient and the three-dimensional operator defined on the implicit two-dimensional manifold, respectively, n. τΓ =n Γ *τ Γ It is a cross-section perpendicular to Γ. The outward unit normal vector, n Γ and τ Γ Representing the Γ normal and The threshold projection of the tangential unit vector and the filter design variable is expressed as: Where β and ξ are the parameters of the threshold projection, which are obtained through numerical calculation and testing; The variational form of the surface partial differential equation filter is defined in a first-order Sobolev space on the surface Γ, obtained according to the Galerkin method: in, It is γ f trial function, Let be a first-order Sobolev space defined on ∑. Let denote a second-order Lebesque space defined on Γ.

5. The fiber bundle topology optimization method for surface flow according to claim 4, characterized in that, The coupling of the design variables is defined on the implicit two-dimensional manifold for the surface flow model, and the coupling relationship is achieved by transforming the tangential gradient operator. Tangential divergence operator div Γ and unit normal vector n Γ This is obtained from the form defined on the bottom flow type ∑; Tangential gradient operator The conversion is based on the following formula: Among them, P Γ It is the projection defined on the tangential space of Γ0, with the normal unit vector n. Γ It can be represented in the following form: Where ||·||² is the second-order modulus of the vector, and the design variable d is used for filtering. f The superscript distinguishes the transformed unit normal vector from its original form. This identification method is then applied to other transformed operators and variables, where the projection P... Γ This is then expressed in the following form: Where I is the second-order unit tensor, and the superscript T indicates the transpose of the tensor or vector; Tangential gradient operator The conversion is based on the following formula: Based on the transformation method of the tangential gradient algorithm, the tangential divergence operator div Γ Convert to Where tr is the rank of the tensor; Due to the tangential gradient operator Depends on d f d f The first variational form of is as follows, for For d f div operator Γ The first-order form is as follows, for Because of d f If the differential is topologically identical, then the Riemannian metric is introduced, which satisfies the differential on the base manifold and the differential on the implicit two-dimensional manifold; in, and Let Γ and ∑ be the differentials of the boundary curves, respectively; Based on the transformation form of the tangential gradient operator and Same topology, tangential gradient operator The transformation is based on the variational form of the filter derived from the partial differential equation of the surface, which yields the coupling of the design variables. gradient operator on Γ Transformed into form.

6. The fiber bundle topology optimization method for surface flow according to claim 5, characterized in that, The surface Navier-Stokes equations on implicit two-dimensional manifolds are as follows: Newton's surface fluid motion equations, which describe two-dimensional manifolds with a codimensional of 1 in Euclidean space, are derived from the surface Navier-Stokes equations based on the laws of conservation of momentum and mass to describe incompressible surface flows: Where u is the fluid velocity, p is the fluid pressure, ρ is the fluid density, and η is the shear viscosity coefficient of the fluid on the implicit two-dimensional manifold, u·n Γ =0 is the tangential constraint on the fluid velocity. The tangential constraint is passively applied because the fluid flows in space on the two-dimensional manifold f, and the fluid velocity is a vector in the tangential space of Γ. Solving the Navier-Stokes equations for the surface is possible because the fluid velocity and pressure at the interface, boundary, and points on the two-dimensional manifold Γ are known. in, The velocity distribution is known and depends on the boundary or interface on the surface ∑. The known flow velocity l at the location v，∑ , l v，Γ It is the boundary on the Γ plane, satisfying At the same time, satisfy l v，Γ It is the interface on the Γ surface, l s，Γ It is a boundary curve with open boundary conditions, and satisfies p 0，Γ The fluid pressure is known and depends on a finite set of points on the surface ∑. The specific fluid pressure p 0，∑ ; A finite set of points; It is a finite set of points; In the above equation, when the known flow velocity is 0, the boundary conditions at the inlet or interface of the flow velocity degenerate into no-slip boundary conditions: Among them, Down, It is the non-slip part of the curved surface; The variational form of the surface Navier-Stokes equations does not consider the function space without tangential constraints on fluid velocity. These tangential constraints are obtained using the Lagrange multiplier method. According to the Galerkin method, the variational form of the surface Navier-Stokes equations is: Where λ is a Lagrange multiplier that imposes a tangential constraint on the velocity; and It is a trial function of u, p, and λ; dΓ and dl r These are the differential and boundary curve of the two-dimensional manifold, respectively. The Lagrange multipliers in the above equation are used to provide constraints on the tangential velocity of the fluid, which is equivalent to uniformly distributing a force field in the normal direction of the two-dimensional manifold. This force field cancels out the centrifugal force, Coriolis force, and Euler force from the fluid particles in the normal direction of the two-dimensional manifold to satisfy the tangential constraint. Because l v，Γ With l v，∑ They are topologically identical. and They are topologically identical, l v，∑ and It is certain. and p 0，Γ respectively with p 0，Σ Since it is topologically identical, based on the coupling relationship of the design variables, the variational form of the above equation is transformed into a form defined on the two-dimensional manifold ∑:

7. The fiber bundle topology optimization method for surface flow according to claim 6, characterized in that, The design objective for topology optimization of surface flow fiber bundles can be expressed in the following universal form: Where A and B are the integrands of the design objective J, and based on the coupling relationship of the design variables, the design objective in the above equation is transformed into a definition on the base manifold ∑: In the above formula, the unit tangential vector τ Γ exist The relationship above: The design goal then becomes: Based on the transformed design objectives, an adjoint analysis of fiber bundle topology optimization was performed on the functional space defined on the bottom manifold.

8. The fiber bundle topology optimization method for surface flow according to claim 7, characterized in that, The bottom manifold, together with the implicit two-dimensional manifold and the surface flow model, forms a fiber bundle, where ∑ is the bottom manifold, Γ×[0,1] are the fibers in the fiber bundle, and proj1 is the natural projection: proj1: ∑×(Γ×[0,1])→∑ satisfies proj1(X ∑ , (X Γ , Γ p ))=proj1(X ∑ , (d f (X ∑ ), γ p ))=X ∑ ,for φ1 is a model with the same topology, φ1: ∑→Γ×[0,1] satisfying φ1(X ∑ )=(X ∑ γ p )=(d f (X ∑ ), γ p For a model with the same topology φ2: Γ×[0,1]→∑×(Γ×[0,1], satisfying φ2(X Γ γ p )=(X ∑ (X ∑ γ p ))=(X ∑ , (d f (X ∑ ), γ p )),for Based on the above representation, the fiber bundle topology optimization for surface flow is constructed as follows: Maximize or minimize Under the following constraints: Where J0 is the initial value of the design variable J; area constraints and volume constraints are applied to the surface flow model and the implicit two-dimensional manifold, respectively; s is the area ratio of the surface flow model; v is the volume ratio of the regions enclosed by the implicit two-dimensional manifold and the bottom flow model; and S0∈(0,1) and v0∈(0,1) are specific area and volume ratios, respectively.

9. The fiber bundle topology optimization method for surface flow according to claim 8, characterized in that, The adjoint analysis for fiber bundle topology optimization is solved using an iterative procedure based on gradient information. The adjoint sensitivity is used to determine the relevant gradient information, which is obtained by adjoint analysis of the design objective, area constraint, and volume constraint. Based on the continuous adjoint analysis method, the evolution process of the sensitivity J of the adjoint design target is as follows: Where, γ fa With d fa These are the filter design variables γ f With d f The adjoint variable is δ, which is a first-order variational operator. The adjoint variable is derived from the adjoint equation in the variational formula. The variational formula for the adjoint equation of the Naver-Stokes equations on surfaces evolves as follows: Among them, u a p a , λ a These are the accompanying variables of u, p, and λ, respectively; They are u a p a , λ a The test Functions; γ and d m It is derived from the adjoint equation of surface partial differential filtering, and its variational formula is as follows: γ fa It belongs to a first-order Sobolev space defined on Σ, such that Find d fa It belongs to a first-order Sobolev space defined on ∑, such that in, and They are γ fa With d fa trial functions; For area constraints, the evolution of the associated sensitivity s is as follows: Based on the above equation, the accompanying sensitivity δ(s|Γ|) changes from s|Γ|=∫ Γ γ p The adjoint analysis of dΓ evolved as follows: In the above equation, by solving for γ and d m The variational formula of the adjoint equation of the surface partial differential filter can be used to obtain the adjoint variable γ. fa With d m : γ fa It belongs to a first-order Sobolev space defined on ∑, such that d fa It belongs to a first-order Sobolev space defined on ∑, such that Based on the evolution of the accompanying sensitivity s, the accompanying sensitivity δ|Γ| changes from |Γ|=∫ Γ The association analysis of 1dΓ evolved from this; In the following equation, by solving for γ and d m The variational formula of the adjoint equation of the surface partial differential filter can be used to obtain the adjoint variable γ. fa With d m : γ fa It belongs to a first-order Sobolev space defined on Σ, such that d fa It belongs to a first-order Sobolev space defined on ∑, such that For volume constraints, the evolution of the associated sensitivity v is as follows: In the above formula, by solving d m The variational formula of the adjoint equation of the surface partial differential filter yields the adjoint variable d. f Accompany variable d fa The variational formula for solving the adjoint equation of surface partial differential filtering is d. m Evolved from: in, d fa It belongs to a first-order Sobolev space defined on ∑. By evolving the design variables using adjoint sensitivity, we obtain γ and d. m The process involves iterative evolution to determine the fiber bundle structure.

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