Explicit shape optimization method for complex shell structures based on embedded spline components

Abstract

The present invention belongs to the technical field of structural optimization design, and specifically relates to an explicit morphology optimization method for complex shell structures based on embedded spline components, comprising steps S1: processing a given complex shell structure using a computational conformal mapping technique to obtain a mapping process, an inverse process, and a parameter domain; S2: arranging morphology components according to a design variable vector; S3: constructing a plane morphology field function based on a topological description function method; S4: forming an embedded morphology description on a curved surface based on an inverse mapping process; S5: constructing an optimization formula and a solution framework based on a specific design problem, submitting the structure for numerical analysis, and extracting the calculation results; S6: performing sensitivity analysis and solving the velocity field; S7: judging whether the optimization process has reached convergence based on the sensitivity and the objective function; and S8: stopping the iteration and further checking the indicators. The present invention efficiently parameterizes the morphology function based on a novel spline component description, enabling the description of complex curved surface morphologies with fewer design variables.

Description

Technical Field

[0001] The present invention belongs to the technical field of structural optimization design, and in particular relates to an explicit morphology optimization method for a complex shell structure based on embedded spline components. Background Art

[0002] Shell structures are among the most common and efficient structural elements in nature and engineering, encompassing applications such as cell walls, microcapsules, biomimetic designs, bridge construction, and aerospace. The superior performance of shell structures stems primarily from their slender, curved geometry, which enables them to efficiently withstand external loads while providing effective protection. However, in practical applications, shell structures are often subjected to extreme service environments and complex loading conditions. Therefore, research on shell structure design has long been a significant focus, primarily categorized into four categories: size optimization, shape optimization, topology optimization, and reinforcement optimization.

[0003] Shell shape optimization is straightforward and involves adjusting specific geometric dimensions such as height, width, and thickness. Shape optimization encompasses two types of techniques:

[0004] On the one hand, CAD-based methods use spline models, especially NURBS expressions used to describe free-form surfaces, to geometrically model shell structures. Here, the corresponding optimization design variables are the parameters used in modeling. Unlike size optimization, the design object of CAD-based shape optimization methods controls the shape but does not represent specific geometric dimensions.

[0005] On the other hand, node-based methods (also called vertex-based / non-CAD-based methods) use the node coordinates of the finite element discretization as design variables. Research on shell structure topology optimization can be traced back to the pioneering work of Cheng and Olhoff. Since then, various powerful methods have been applied to this problem, including SIMP (Solid Isotropic Material with Penalization) method, BESO (Bi-directional Evolutionary Structural Optimization) method, LSM (Level Set Method) method, and MMC (Moving Morphable Components) method.

[0006] Among them, thanks to the fact that the design object is given specific geometric meaning, the MMC method can obtain the final design with a clear force transmission path with fewer design variables.

[0007] Unlike the aforementioned methods, reinforcement design typically does not directly modify the shell structure itself, but instead focuses on the design of its attached reinforcement ribs. Reinforced shell structures are widely used in various scenarios due to their convenience and efficiency.

[0008] Furthermore, the aforementioned methods are not strictly independent. Many integrated collaborative optimization methods have been proposed to further improve structural performance, such as the joint optimization of the shape and topology of shell structures, the use of topology optimization methods to design stiffeners, and the collaborative optimization of the base plate shape and ribs of stiffener structures.

[0009] For complex and continuously changing structural components, the aforementioned algorithms require higher computational power to calculate the parameterized shape, resulting in an explosive increase in the number of design variables used, which is not conducive to simplifying the explicit shape of complex shell structures. Therefore, we propose an explicit shape optimization method for complex shell structures based on embedded spline components. Summary of the Invention

[0010] The purpose of the present invention is to provide an explicit morphology optimization method for complex shell structures based on embedded spline components, which efficiently parameterizes the morphology function based on a new spline component description, so that complex surface morphology can be described with fewer design variables.

[0011] The technical solutions adopted by the present invention are as follows:

[0012] The explicit shape optimization method of complex shell structure based on embedded spline components includes the following steps:

[0013] S1: For a given complex shell structure, computational conformal mapping technology is used to process it, obtaining the mapping process, inverse process, and parameter domain; the design variable vector is initialized in the parameter domain;

[0014] S2: Arrange the topography components according to the design variable vector;

[0015] S3: Constructing plane topography field function based on topological description function method;

[0016] S4: Based on the inverse mapping process, an embedded topography description is formed on the surface; the surface cutting operation and multi-faceted splicing technology are integrated to form the topography definition of any complex surface;

[0017] S5: Construct optimization formulas and solution frameworks based on specific design problems, submit structural numerical analysis, and extract calculation results;

[0018] S6: Perform sensitivity analysis and solve the velocity field;

[0019] Further calculate the discrete sensitivity and submit it to the optimization solver MMA to update the design variable vector;

[0020] S7: If the optimization process reaches convergence based on the sensitivity and the objective function, then jump to step S8;

[0021] Or if it is determined that the optimization process has not reached convergence, jump to step S2;

[0022] S8: The iteration stops and further indicator verification is performed based on the final design.

[0023] The step S1 comprises:

[0024] Let f:S→M be the computational conformal mapping to be solved, and M is defined in The rectangular parameter domain in , S is the surface of the shell structure, and conformality is achieved by combining two quasi-conformal mappings with appropriate Beltrami coefficients, where:

[0025] The first quasi-conformal mapping f1: is the disk harmonic map obtained by solving the following equations (1a) and (1b) using the finite element method:

[0026]

[0027] Where, Δ S It is the Laplace-Beltrami operator defined on the surface S, and the symbol G represents the operator defined on the complex plane. The target disk area, and represent the boundaries of S and G respectively;

[0028] Secondly, the Linear Beltrami Solver technique is used to eliminate the conformal distortion of the first mapping, and then is the inverse mapping The Beltrami operator, where ρ, τ, and j represent the real part, imaginary part, and imaginary unit of a complex number, respectively;

[0029] The second required quasi-common mapping is f2(x+jy)=u(x,y)+jv(x,y):G→M, where (x,y) and (u,v) are complex coordinates defined on the disk D and the rectangular parameter domain M, respectively. According to the Linear Beltrami Solver technique, the mapping f2 can be reconstructed by solving the following set of partial differential equations (2a), (2b), and (2c):

[0030]

[0031] In the formula And there is here, Represents the boundary of the rectangular parameter domain M, and the final mapping is constructed according to the mapping compound as

[0032] The step S2 uses a new spline component description to efficiently parameterize the topography function, based on the characteristic function φ i The definition of (u,v) is used to deal with the complexity of the high variation in the morphological description and the complexity of the boundary of the perturbation region, as follows:

[0033] First, the local coordinate system (u', v') is constructed using the following equation (3):

[0034]

[0035] Where, and is the coordinate of the center point of the i-th component, θ i is the rotation angle about the global Cartesian coordinate system; secondly, convert the Cartesian coordinates to polar coordinates :

[0036]

[0037]

[0038] Thus, the boundary of the i-th component can be expressed in radial form:

[0039]

[0040] Where, is the kth shape factor of the i-th component, and R k,p is a NURBS basis function of order p, assuming p = 2, and has the following form:

[0041]

[0042] Where w k is the weight corresponding to the kth basis function, assuming w k =1, and is the kth B-spline basis function defined by the Cox-de Boor recursive formula:

[0043]

[0044] And there are:

[0045]

[0046] Where, Belong to the node vector The kth node of

[0047] Finally, the indicator function φ i (u,v) is defined by the normalized form of equation (10):

[0048]

[0049] Where, the symbol q represents the shape index, assuming q = 2 and eps = 10 -15 is a small quantity to avoid singularity. In summary, the i-th component-based design variable vector is defined as The design variable vectors describing the global morphology are assembled as D = (D1, D2, ..., D n ), the topography function is thus defined as h S (x; D) = h M (p; D).

[0050] The step S3 introduces the moving deformable component method to approximate the optimal plane topography function, which is as follows:

[0051] The topological description function is used to characterize the disturbed area. For the i-th component, its topological description function is expressed as:

[0052]

[0053] Where M i is a subset of the parameter domain M, representing the area occupied by the i-th component, and the plane topography function is further constructed as:

[0054]

[0055] In the formula, the symbol is the design variable representing the height of the i-th topographic component, H(·) is the normalized Heaviside function, which maps the independent variable to the interval [0,1]. At the global level, the plane topography function h M It consists of the topography field function of each component:

[0056]

[0057] Where nc represents the total number of components in the design domain.

[0058] In step S4, the surface configuration S is used to represent the mid-surface of the designed shell, and a topographic perturbation is performed on any point x on the surface based on the unit normal vector N(x) at its location, as follows:

[0059] x′(x)=x+h S (x)·N(x),(14)

[0060] Where h S(x) is the surface topography field (STF) function that reflects the amplitude variation. The embedded topography description is used. The mapping relationship f:S→M solved in step S1 is used to map the shell mid-surface to the standard parameter domain. The topography function is defined in the obtained parameter domain. Finally, the inverse topography mapping is embedded into the mid-surface. Specifically, the domain of the STF function is reduced to the rectangular parameter plane M, which is:

[0061] h S (x) = h S (f -1 (p))=h M (p),(15)

[0062] Where f(x) = p(u,v), and h M represents the plane topography function;

[0063] Furthermore, the expression of surface perturbation is adjusted to:

[0064] x′(x)=x+h M (p)·N(x),(16)

[0065] At this time, under the embedded morphology description framework, the construction process of the perturbation amplitude of each point x is simplified from the surface S to the rectangular parameter domain M.

[0066] In step S4, surface cutting operations and multi-faceted splicing techniques are integrated to process general complex surfaces. For example, a cylindrical surface is processed as follows:

[0067] First, cut along the generatrix Γ and obtain the process surface S * ;

[0068] At this time, the surface S * The topology is equivalent to a planar quadrilateral, so the CCM technique can be used to obtain the conformal mapping f:S * →M, where M represents the corresponding rectangular parameter domain, considering the surface S * The construction process of the morphology function follows the aforementioned rules, namely The symbol x * and p represent the surface S * Points on the original surface S, note that the curve Γ is derived as the surface S * The two curves on and

[0069] Therefore, any point x∈Γ is split into two points and This further leads to the topography function h S The incompatibility of the values taken on the curve Γ, returning to the original surface S, the topography function h SIt is thus defined as:

[0070]

[0071] We further introduce the multi-faceted patch stitching technology, which divides the original surface into multiple simple surface patches based on geometric characteristics, and then further define the shape functions, and finally realize the stitching of the surface patches. Assume is a surface segmentation, where S i is the i-th surface patch. Note that there is no additional requirement for the shape of each surface patch. For any surface patch S i , and its morphology field is ;

[0072] In order to define the overall surface shape, the local shape function Extension to the global domain:

[0073]

[0074] Where D i is the corresponding surface patch S i The design variable vector of , by using the definition of continuation, the overall surface morphology function is assembled as:

[0075]

[0076] In step S5, the morphology optimization for minimizing the structural stiffness is considered, the material used in the structure is set to be linear elastic, isotropic and uniformly distributed, and it is assumed that S(D) is the mid-surface of the shell structure disturbance;

[0077] That is, S0=S(0) represents the original mid-surface, represents the area occupied by the perturbed shell structure, where t is the coordinate in the thickness direction, is the thickness of the shell structure;

[0078] At this point, assuming that the surface S is the core consideration of geometric modeling, the main consideration is the solid area corresponding to the shell structure. In the current framework based on spline components, the morphology optimization problem can be formulated as follows:

[0079] Find D,U(x;D) (20a)

[0080]

[0081]

[0082]

[0083]

[0084] In the formula, symbols C and F(D) represent the structural flexibility function and external force respectively. and is the constitutive tensor defined in the curved coordinate system. In equation (20d), the real displacement U(x; D) belongs to the preset constraint set U, and the virtual displacement V(x; D) belongs to the feasible set U. ad , both displacements satisfy the Reissner-Mindlin kinematic assumption; in addition, e αβ (·) is the surface strain tensor, e α3 (·) is the transverse shear strain tensor, and It is a set used to avoid infeasible descriptions.

[0085] The integration domain of the structural response in step S6 changes with the iteration process, so the shape sensitivity analysis method is used to deal with the shape change;

[0086] For simplicity of presentation, the symbol d is used here to represent the parameter controlling the shell shape;

[0087] Assume I(W) = ∫ B(d,t) WdV is a generalized structural response that meets the normative requirements of the design domain and has sufficient smoothness, where W = W(U(x; d), V(x; d)) is the functional of U(x; d) and V(x; d), where V(x; d) and V(x; d) are the displacements corresponding to U(x; D) and V(x; D), respectively.

[0088] According to Raynold's transport theorem, the material time derivative of I(W) is:

[0089]

[0090] Where div(·) is the divergence operator, and symbol v is the velocity field induced by the instantaneous change of parameter d. By using Gauss’s theorem, the second term in equation (21) is reduced to the boundary :

[0091]

[0092] Among them, V n =v·N and N represents the boundary defined The outward normal on , and:

[0093]

[0094] Based on the above conclusions, the material time derivative of the flexibility function for:

[0095]

[0096] because (like Figure 8 (a), then:

[0097]

[0098] And there are:

[0099]

[0100] For any point x'=x'(x; d) on the mid-surface S(d), its position is obtained by the perturbation operation of the point x on the original surface S0, that is:

[0101] x'(x;d)=x+h S (x;d)·N(x,S0), (27)

[0102] Where N(x, S0) is the positive normal vector of surface S0 at point x. Projecting point x' along the local normal vector to the upper surface Γ1 and the lower surface Γ2, the corresponding projection point is expressed as:

[0103]

[0104] Where N(x',S(d)) represents the positive normal vector of the surface S(d) at point x'.

[0105] In step S6, the velocity field induced by the instantaneous change of the parameter d corresponding to the upper boundary Γ1 is derived, and equation (27) is substituted into (28a) to obtain:

[0106]

[0107] The velocity field v1 of the surface Γ1 is defined as the material time derivative of x' with respect to d, that is:

[0108]

[0109] And there are:

[0110]

[0111] Where N(x'1, Γ1) is the outer normal at point x'1 on the boundary Γ1. Considering that the topography field is defined by surface cutting operation and multi-faceted patch splicing technology, h S The derivative of (x; d) with respect to d can be written as:

[0112]

[0113] Where, Represents the definition on the surface patch S i The morphology field on Represents its extended definition in the overall structure, and corresponds to S i The process surface, here, pass Get, and have Represents the i-th conformal mapping, function The partial derivative with respect to d is calculated as follows:

[0114]

[0115] Among them, h i,k (p; d) is the topography field function defined for the kth component in the ith subdomain.

[0116] The technical effects achieved by the present invention are:

[0117] In the present invention, the explicit morphology optimization method of complex shell structures based on embedded spline components is proposed. The complex and continuously changing morphology is composed of moving, deforming and covering structural components, which makes the proposed algorithm have the advantages of clear and brilliant structural boundaries and interpretable final design.

[0118] The present invention provides an explicit shape optimization method for complex shell structures based on embedded spline components. It uses a novel spline component description to efficiently parameterize the shape function, thereby enabling the description of complex surface shapes with fewer design variables.

[0119] The method for explicitly optimizing the shape of a complex shell structure based on embedded spline components of the present invention can easily depict the complex shape of a disturbance region thanks to the strong deformation capability of NURBS.

[0120] The present invention provides an explicit shape optimization method for complex shell structures based on embedded spline components. The local support of the spline basis function avoids the risk of the Runge-Kutta phenomenon in shape interpolation; further, this also ensures the numerical stability of the optimization process.

[0121] The present invention's explicit morphology optimization method for complex shell structures based on embedded spline components adopts computational conformal mapping technology to process the shell structure geometry, integrates surface segmentation operations and multi-facet splicing strategies, and greatly expands the scope of application of the optimization method. BRIEF DESCRIPTION OF THE DRAWINGS

[0122] Figure 1 This is a flow chart of an explicit morphology optimization method for a complex shell structure based on embedded spline components according to a first embodiment of the present invention.

[0123] Figure 2 Schematic diagram of solving a conformal mapping by combining two quasi-conformal mappings according to the first embodiment of the present invention; wherein (a) is the first mapping diagram; (b) is the second mapping diagram;

[0124] Figure 3 Schematic diagram of a spline component with eight shape coefficients according to the first embodiment of the present invention;

[0125] Figure 4 Schematic diagram of the construction process of the PTF function of Example 1 of the present invention, wherein (a) is the TDF function representing the perturbation domain, (b) is the use of the Heaviside function to regularize the height value, (c) is the scaling of the shape size according to the height design variable, and (d) is the construction of the global planar shape function through the shape assembly process;

[0126] Figure 5 is an explanatory diagram of the morphological changes of Example 1 of the present invention;

[0127] Figure 6 Schematic diagram of the implementation process of embedded topography description according to the first embodiment of the present invention, wherein (a) indicates the first step of solving the conformal mapping and obtaining the parameter domain using the CCM technique, (b) indicates designing the topography change in the parameter domain, and (c) indicates embedding the topography design into the original surface.

[0128] Figure 7 Schematic diagram of defining a surface topography field function through a surface cutting operation in accordance with the first embodiment of the present invention;

[0129] Figure 8 Schematic diagram of a first embodiment of the present invention for defining a surface topography field function using a multi-facet splicing technique, wherein (a) represents dividing the original surface into four blocks based on geometric features, (b) represents constructing the topography of each facet using the aforementioned method, including embedding description and surface cutting operations, and (c) represents the topography of the entire surface obtained by splicing all facets with the corresponding topography.

[0130] Figure 9 Schematic diagram of the surface-body relationship of the first embodiment of the present invention, wherein (a) shows the relationship between the mid-curved surface and the body domain, (b) shows the decomposition of the body domain boundary into three parts, (c) shows that the thickness and curvature cause the velocity fields of the upper and lower surfaces to be inconsistent, so the points on these two surfaces need to be considered separately, and (d) shows the body domain at a discrete level;

[0131] Figure 10 Schematic diagram of the problem setting for the conical shell example in Example 2 of the present invention, where (a) shows the reconstruction of the geometric model from 3D point cloud data, and (b) shows the shell structure fixed at the four corners with a thickness of 3 and subjected to five concentrated forces (four at the middle of the edges and one at the center of the surface);

[0132] Figure 11Schematic diagram of the iteration history and intermediate results of Example 2 of the present invention, where (a) represents the initial design (C=424.74) and (b) represents the final design (C=109.42);

[0133] Figure 12 Schematic diagram of the iterative history and intermediate results of the shape-topography joint design in the second embodiment of the present invention. DETAILED DESCRIPTION

[0134] In order to make the purpose and advantages of the present invention more clearly understood, the present invention is described in detail below with reference to the following examples. It should be understood that the following text is only used to describe one or more specific embodiments of the present invention and does not strictly limit the scope of protection of the present invention.

[0135] Example 1:

[0136] like Figure 1-9 As shown in FIG, the explicit morphology optimization method of a complex shell structure based on embedded spline components includes the following steps:

[0137] S1: For a given complex shell structure, computational conformal mapping technology is used to process it, obtaining the mapping process, inverse process, and parameter domain; the design variable vector is initialized in the parameter domain;

[0138] S2: Arrange the topography components according to the design variable vector;

[0139] S3: Constructing plane topography field function based on topological description function method;

[0140] S4: Based on the inverse mapping process, an embedded topography description is formed on the surface; the surface cutting operation and multi-faceted splicing technology are integrated to form the topography definition of any complex surface;

[0141] S5: Construct optimization formulas and solution frameworks based on specific design problems, submit structural numerical analysis, and extract calculation results;

[0142] S6: Perform sensitivity analysis and solve the velocity field; further calculate the discrete sensitivity and submit it to the optimization solver MMA to update the design variable vector;

[0143] S7: Determine whether the optimization process has reached convergence based on the sensitivity and the objective function. If it has converged, jump to step S8; if not, jump to step S2;

[0144] S8: The iteration stops and further indicator verification is performed based on the final design.

[0145] Step S1 specifically includes:

[0146] Let f:S→M be the computational conformal mapping to be solved, and M is defined in The rectangular parameter domain in (here, S is the surface of the shell structure); Different from the direct calculation of the mapping, we achieve conformality by combining two quasi-conformal mappings with appropriate Beltrami coefficients, which greatly improves the computational efficiency;

[0147] The first quasi-conformal mapping f1: It is the disk harmonic map obtained by solving the following equation using the finite element method:

[0148]

[0149] Where, Δ S It is the Laplace-Beltrami operator defined on the surface S, and the symbol G represents the operator defined on the complex plane. The target disk area (see Figure 2 (a)), and represent the boundaries of S and G respectively;

[0150] Second, we use the Linear Beltrami Solver (LBS) technique to eliminate the conformal distortion of the first mapping, so that is the inverse mapping The Beltrami operator, where ρ, τ, and j represent the real part, imaginary part, and imaginary unit of a complex number, respectively;

[0151] Let f2(x+jy)=u(x,y)+jv(x,y):G→M be the second required quasi-universal mapping (see Figure 2 (b)), where (x, y) and (u, v) are complex coordinates defined on the disk D and the rectangular parameter domain M, respectively;

[0152] According to the LBS technique, the mapping f2 can be reconstructed by solving (also using the finite element method) the following set of partial differential equations:

[0153]

[0154] In the formula And there is here, Represents the boundary of the rectangular parameter domain M;

[0155] The final mapping is constructed according to the mapping compound

[0156] Step S2 specifically includes:

[0157] A new spline component description is used to efficiently parameterize the shape function based on the characteristic function φ i(u,v) is defined to handle the complexity of the high variation in the morphological description and the complexity of the boundary of the disturbance region; specifically, Figure 3 As shown:

[0158] First, construct the local coordinate system (u', v') using the following formula:

[0159]

[0160] Where, and is the coordinate of the center point of the i-th component, θ i is the rotation angle about the global Cartesian coordinate system. These design variables give the component the ability to move as a rigid body;

[0161] Next, we convert the Cartesian coordinates to polar coordinates :

[0162]

[0163]

[0164] Thus, the boundary of the i-th component can be expressed in radial form:

[0165]

[0166] Where, is the kth shape coefficient of the i-th component ( So that the boundary forms a closed loop), and R k,p is a NURBS basis function of order p (p = 2 in this study) and has the following form:

[0167]

[0168] In the above formula, w k is the weight corresponding to the kth basis function (in this study w k =1), and is the kth B-spline basis function defined by the Cox-de Boor recursive formula:

[0169]

[0170] And there are:

[0171]

[0172] Where, Belong to the node vector The kth node of

[0173] Finally, the indicator function φ i(u,v) is defined by the following normalized form:

[0174]

[0175] Here, the symbol q represents the shape index (q = 2 in this study), and eps = 10 -15 It is a small amount that avoids singularity;

[0176] In summary, the i-th component-based design variable vector is defined as The design variable vectors describing the global morphology are assembled as D = (D1, D2, ..., D n ), the topography function is thus defined as h S (x; D) = h M (p; D).

[0177] Step S3 specifically includes:

[0178] The moving deformable component method is introduced to approximate the optimal plane topography function. Specifically, the topology description function (TDF) is used to characterize the perturbed area.

[0179] For the i-th component, its topological description function is expressed as:

[0180]

[0181] Where M i is a subset of the parameter domain M, representing the area occupied by the i-th component (e.g. Figure 4 (a) shows), note that the topological description function only indicates whether the region is disturbed and loses height information;

[0182] Therefore, if Figure 4 As shown in (bc), the topography function is further constructed as:

[0183]

[0184] In the formula, the symbol is the design variable representing the height of the i-th topographic component, H(·) is the normalized Heaviside function, which maps the independent variable to the interval [0,1]. At the global level (e.g. Figure 4 (d)), the plane topography function h M It consists of the topography field function of each component:

[0185] h M =max(h1,…,h i ,…,h nc ), (13)

[0186] Where nc represents the total number of components in the design domain.

[0187] Step S4 specifically includes:

[0188] The surface configuration S is used to represent the mid-surface of the designed shell. The topography perturbation of any point x on the surface is performed based on the unit normal vector N(x) at its location, as follows:

[0189] x′(x)=x+h S (x)·N(x),(14)

[0190] Where h S (x) is the surface topography field (STF) function that reflects the amplitude of the change, such as Figure 5 As shown, constructing such a point function involves complex surface operations and will cause problems when extending to arbitrarily complex surface shell models. In this context, we adopt an embedded morphology description and use the mapping relationship f:S→M solved in S1 to map the shell mid-surface to the standard parameter domain. In the obtained parameter domain, the morphology function is defined and finally the inverse morphology mapping is embedded into the mid-surface.

[0191] Specifically, the domain of the STF function is reduced to the parameter plane M, that is:

[0192] h S (x) = h S (f -1 (p))=h M (p), (15)

[0193] Where f(x) = p(u,v), and h M represents the plane topography field (PTF) function. Furthermore, the expression of the surface perturbation is adjusted to

[0194] x′(x)=x+h M (p)·N(x), (16)

[0195] This means that in the embedded topography description framework, the construction process of the perturbation amplitude at each point x is simplified from the surface S to the parameter domain M, as follows: Figure 6 As shown;

[0196] In addition, we integrate surface cutting operations and multi-faceted splicing technology to handle general complex surfaces. Figure 7 Taking the cylindrical surface shown as an example, we first cut along the generatrix Γ and obtain the process surface S * ;

[0197] At this time, the surface S *The topology is equivalent to a planar quadrilateral, so the CCM technique can be used to obtain the conformal mapping f:S * →M, where M represents the corresponding parameter domain;

[0198] Considering the surface S * The construction process of the morphology function follows the aforementioned rules, namely The symbol x * and p represent the surface S * Points on the original surface S, note that the curve Γ is derived as the surface S * The two curves on and

[0199] Therefore, any point x∈Γ is split into two points and This further leads to the topography function h S The incompatibility of the values taken on the curve Γ, returning to the original surface S, the topography function h S It is thus defined as:

[0200]

[0201] Considering that the single mapping process of the aforementioned method may introduce numerical instability due to its global characteristics;

[0202] Therefore, the multi-faceted patch splicing technology is further introduced to deal with this problem. The original surface is divided into multiple simple surface patches based on geometric characteristics, and then the shape functions are further defined respectively. Finally, the surface patches are spliced together, such as Figure 8 As shown;

[0203] Similar to surface cutting technology, patch segmentation also requires further special processing;

[0204] Assumptions is a surface segmentation, where S i is the i-th surface patch. Note that there is no additional requirement for the shape of each surface patch. For any surface patch S i , and its morphology field is ;

[0205] In order to define the overall surface shape, the local shape function Extension to the global domain:

[0206]

[0207] Where D i is the corresponding surface patch S i The design variable vector of

[0208] By using the definition of continuation, the topography function of the entire surface is assembled as:

[0209]

[0210] Step S5 specifically includes:

[0211] In this paper, we consider the topography optimization problem of minimizing the stiffness of a structure, where the material used is assumed to be linear elastic, isotropic, and uniformly distributed.

[0212] Assuming S(D) is the mid-surface of the shell structure perturbation, we have S0=S(0) representing the original mid-surface, represents the area occupied by the perturbed shell structure, where t is the coordinate in the thickness direction, is the thickness of the shell structure;

[0213] Unlike the previous chapters, where the surface S is the core consideration for geometric modeling, this chapter mainly considers the solid area corresponding to the shell structure (see Figure 9 (a));

[0214] In the current framework based on spline components, the shape optimization problem can be formulated as:

[0215] Find D,u(x;D) (20a)

[0216]

[0217]

[0218]

[0219] In the formula, symbols C and F(D) represent the structural flexibility function and external force respectively. and is the constitutive tensor defined in the curved coordinate system;

[0220] In equation (20d), the real displacement U(x; D) belongs to the preset constraint set U, and the virtual displacement V(x; D) belongs to the feasible set U ad ;Both displacements satisfy the Reissner-Mindlin kinematic assumption;

[0221] In addition, e αβ (·) is the surface strain tensor (including membrane behavior and bending behavior), e α3 (·) is the transverse shear strain tensor, and It is a set used to avoid infeasible descriptions.

[0222] Step S6 specifically includes:

[0223] The integration domain of the structural response changes with the iterative process, so we use the shape sensitivity analysis method to deal with the shape changes;

[0224] For simplicity of presentation, the symbol d is used here to represent the parameter that controls the shape of the shell;

[0225] Assume I(W) = ∫ B(d,t) WdV is a generalized structural response (assuming that I(W) meets the normative requirements of the design domain and has sufficient smoothness), where W = W(U(x; d), V(x; d)) is the functional of U(x; d) and V(x; d), where V(x; d) and V(x; d) are the displacements corresponding to U(x; D) and V(x; D), respectively;

[0226] According to Raynold's transport theorem, the material time derivative of I(W) is:

[0227]

[0228] Where div(·) is the divergence operator, and the symbol v is the velocity field induced by the instantaneous change of the parameter d;

[0229] By using Gauss's theorem, the second term in Eq. (21) is reduced to the boundary

[0230]

[0231] Among them, V n =v·N and N represents the boundary defined The outward normal on , and:

[0232]

[0233] Based on the above conclusions, the material time derivative of the flexibility function for:

[0234]

[0235] because (like Figure 9 (a), we have:

[0236]

[0237] And there are:

[0238]

[0239] like Figure 9As shown in (b), for any point x'=x'(x; d) on the mid-surface S(d), its position is obtained by the perturbation operation of the point x on the original surface S0, that is:

[0240] x'(x;d)=x+h S (x;d)·N(x,S0), (27)

[0241] Where N(x, S0) is the positive normal vector of surface S0 at point x, and point x' is projected along the local normal vector to the upper surface Γ1 and the lower surface Γ2 (e.g. Figure 9 (c)), the corresponding projection point is expressed as:

[0242]

[0243] Where N(x', S(d)) represents the positive normal vector of the surface S(d) at point x'. Next, we take the upper boundary Γ1 as an example to derive the corresponding velocity field induced by the instantaneous change of the parameter d. Substituting equation (27) into (28a), we obtain:

[0244]

[0245] The velocity field v1 of the surface Γ1 is defined as the material time derivative of x' with respect to d, that is:

[0246]

[0247] And there are:

[0248]

[0249] Where N(x'1, Γ1) is the external normal at point x'1 on boundary Γ1. The derivation of the velocity field on boundary Γ2 is very similar to the above process and will not be repeated here.

[0250] In addition, due to the slender nature of the structure, the integral on the boundary Γ3 is neglected;

[0251] Considering that the topography field is defined by surface cutting operations and multi-faceted patch stitching techniques, h S The derivative of (x; d) with respect to d can be written as:

[0252]

[0253] Where, Represents the definition on the surface patch S i The morphology field on Represents its extended definition in the overall structure, and corresponds to S i The process surface of pass Get, and have represents the i-th conformal mapping;

[0254] function The partial derivative with respect to d is calculated as follows:

[0255]

[0256] Among them, h i,k (p; d) is the topography field function defined for the kth component in the ith subdomain.

[0257] Example 2:

[0258] like Figure 10-12 As shown in Figure 2, we use point clouds (through 3D scanning technology) to obtain a reconstructed dome shell structure to verify the applicability of the existing optimization method in actual engineering structures. The geometric dimensions and boundary conditions are as follows: Figure 10 shown.

[0259] First, we implemented the morphology design, the iterative process and the intermediate results are as follows Figure 11 As shown in the figure, the optimization results converge quickly with the designed structure. The final design has raised corners, which improves the local stiffness near the support points.

[0260] Then, we performed a shape-topography joint design, e.g. Figure 12 As shown. Before step 300, the iterative process uses the shape design method to optimize the structural performance, where the flexibility function is reduced from 629.80 to 210.08. In step 301, the proposed terrain design method is used to further improve the structural performance; the base surface is the synthetic surface of the shape optimization process, and the flexibility function is further simplified to 61.62. It can be seen from the curve that the shape optimization further improves the shape optimization results. The experimental results show that compared with the shell shape design alone ( Figure 12 300th step result in ) or individually designed morphology ( Figure 11 Compared with (b), the joint optimization can obtain a stronger structure. In addition, the joint optimization results are significantly different from the independent morphology design results, which shows that the morphology design is more sensitive to the curvature of the substrate surface.

[0261] The foregoing is merely a preferred embodiment of the present invention. It should be noted that those skilled in the art may make various improvements and modifications without departing from the principles of the present invention, and such improvements and modifications are also within the scope of protection of the present invention. Structures, devices, and operating methods not specifically described or explained herein shall, unless otherwise specified or limited, be implemented in accordance with conventional means in the art.

Claims

1. An explicit topography optimization method for complex shell structures based on embedded spline components, characterized by: The following steps are involved: S1: For a given complex shell structure, computational conformal mapping technology is used to process it, obtaining the mapping process, inverse process, and parameter domain; the design variable vector is initialized in the parameter domain; S2: Arrange the topography components according to the design variable vector; S3: Constructing plane topography field function based on topological description function method; S4: Based on the inverse mapping process, an embedded topography description is formed on the surface; the surface cutting operation and multi-faceted splicing technology are integrated to form the topography definition of any complex surface; S5: Construct optimization formula and solution framework based on design problem, submit structural numerical analysis and extract calculation results; In step S5, the morphology optimization is considered to minimize the structural stiffness, the material used in the structure is set to be linear elastic, isotropic and uniformly distributed, and S(D) is set as the mid-surface of the shell structure disturbance; That is, S0=S(0) represents the original mid-surface, represents the area occupied by the perturbed shell structure, where t is the coordinate in the thickness direction, is the thickness of the shell structure. At this time, let the surface S be the core consideration of geometric modeling. The solid area corresponding to the shell structure is used. The morphology optimization problem can be listed as follows: Find D,U(x;D) (20a) Minimize C=C(U(x;D),D)=∫ B(D,t) (F(D)·U(x;D))dV, (20b) In the formula, symbols C and F(D) represent the structural flexibility function and external force respectively. and is the constitutive tensor defined in the curved coordinate system. In equation (20d), the real displacement U(x; D) belongs to the preset constraint set U, and the virtual displacement V(x; D) belongs to the feasible set U. ad , both displacements satisfy the Reissner-Mindlin kinematic theory; in addition, e αβ (·) is the surface strain tensor, e α3 (·) is the transverse shear strain tensor, and It is a set used to avoid infeasible description of the shape; S6: Perform sensitivity analysis and solve the velocity field; Calculate the discrete sensitivity and submit it to the optimization solver MMA to update the design variable vector; S7: If the optimization process reaches convergence based on the sensitivity and the objective function, then jump to step S8; Or if it is determined that the optimization process has not reached convergence, jump to step S2; S8: The iteration stops and the indicators are checked according to the final design.

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