Multi-plate isogeometric topological optimization-based porous reinforced shell full-scale design method

Abstract

The invention belongs to the technical field related to structure optimization, and discloses a porous reinforced shell full-scale design method based on multi-piece isogeometric topology optimization, which comprises the following steps: (1) respectively dividing a thin shell structure and a reinforcing rib area into high-order NURBS unit grids; (2) an isogeometric Kirchhoff-Love shell unit is adopted to describe a thin shell structure, a three-dimensional entity unit is adopted to describe a reinforcing rib area, a high-order NURBS unit grid is decomposed into Bezier units, and then an entity-shell coupling unit is constructed; and (3) setting an equivalent density on each entity unit as a design variable, converting the design variable into the equivalent density on a thin shell control point by applying a mapping constraint, introducing a local volume constraint to the design variable so as to obtain a bone-shaped porous reinforcing rib, and further constructing a full-scale multi-piece isogeometric topological optimization model for the porous reinforced shell. And the finally optimized porous reinforced shell structure is further obtained. According to the invention, the applicability is improved.

Description

Technical Field

[0001] The present invention belongs to the technical field related to structural optimization, and more specifically, relates to a full-scale design method for porous stiffened shells based on multi-slice isogeometric topology optimization. Background Art

[0002] The stiffened shell structure can achieve properties such as light weight, high strength, and high robustness by cleverly arranging stiffeners on a thin shell, and plays a key role in the aerospace, ship, and automotive industries. Topology optimization is a structural design method that automatically finds the optimal material distribution within the design domain according to artificially specified objective functions and constraint conditions. Using the topology optimization method to optimize the stiffened shell structure can fully exploit the performance potential of the stiffened shell structure. At the same time, isogeometric analysis is an analysis method that directly uses the shape functions of the geometric model to construct the analysis grid, and has the characteristics of being convenient for constructing high-order continuous grids and high analysis accuracy. Introducing isogeometric analysis when performing topology optimization on the stiffened shell structure can further improve the analysis and optimization accuracy. Therefore, in order to meet the engineering needs and promote the lightweight design of structures, it is necessary to develop an efficient and flexible isogeometric topology optimization design method for stiffened shells.

[0003] Regarding the isogeometric topology optimization method for stiffened shells, relevant technical personnel in this field have conducted some research. For example, in Document 1: "Sun Y, Zhou Y, Ke Z, et al. Stiffener layout optimization framework by isogeometric analysis-based stiffness spreading method[J]. Computer Methods in Applied Mechanics and Engineering, 2022, 390: 114348.", a stiffener layout optimization method based on the stiffness spreading method of isogeometric analysis is proposed. This method can optimize the layout and size of stiffeners simultaneously. By using beam elements to describe the stiffener structure, the calculation efficiency is high. However, the design freedom is restricted to a certain extent, and the layout of stiffeners can only be optimized on a flat design domain. Regarding the isogeometric topology optimization method for porous stiffened shells, relevant technical personnel in this field have also conducted some research. For example, in Document 2: "Sun Y, Zhou Z, Lai P, et al. Isogeometric analysis-based buckling optimization framework for grid-stiffened shells using asymptotic homogenization method and Rayleigh-Ritz method[J]. Structural and Multidisciplinary Optimization, 2022, 65(11): 330.", a buckling optimization framework for grid-stiffened cylindrical shells based on isogeometric analysis is proposed. This method first designs a grid-stiffened unit cell on a square plate using the asymptotic homogenization method, and then fills the grid-stiffened unit cell onto the cylindrical shell to obtain a grid-stiffened cylindrical shell. This method can design a porous stiffener structure with good design robustness for the cylindrical shell. However, this method limits its design freedom due to only considering the periodic array of grid-stiffened unit cells on the cylindrical shell, and is only limited to the stiffening design of cylindrical shells.

[0004] Therefore, it is a current research hotspot problem to be solved to propose an isogeometric topology optimization method that can be used for the stiffening design of thin-shell structures with complex geometric shapes and obtain a porous stiffened shell with high robustness. Summary of the Invention

[0005] In view of the above defects or improvement requirements of the prior art, the present invention provides a full-scale design method for porous stiffened shells based on multi-patch isogeometric topology optimization, aiming to solve the problem that the existing optimization methods are only limited to the stiffening design of cylindrical shells.

[0006] To achieve the above object, according to one aspect of the present invention, a full-scale design method for a porous stiffened shell based on multi-patch isogeometric topology optimization is provided, and the method includes the following steps:

[0007] (1) First, use a plurality of two-dimensional NURBS patches and a plurality of three-dimensional NURBS patches to construct the geometric model of the thin shell structure and the geometric model of the stiffener region respectively, and divide the thin shell structure and the stiffener region into high-order NURBS element meshes respectively;

[0008] (2) Use isogeometric Kirchhoff–Love shell elements to describe the thin shell structure, use three-dimensional solid elements to describe the stiffener region, introduce the Bézier extraction technology to decompose the high-order NURBS element mesh into a series of C 0 continuous Bézier elements, and then add each element in the shell element stiffness matrix to the element at the corresponding degree of freedom in the corresponding solid element stiffness matrix to construct a solid-shell coupled element. At the same time, introduce displacement penalty terms and rotation penalty terms between multiple NURBS patches respectively;

[0009] (3) Set the equivalent density on each solid element as the design variable, transform the design variable into the equivalent density on the thin shell control points by applying mapping constraints, and introduce local volume constraints on the design variable to obtain bone-like porous stiffeners, and then construct a full-scale multi-patch isogeometric topology optimization model for the porous stiffened shell. The full-scale multi-patch isogeometric topology optimization model is optimized iteratively to obtain an optimized porous stiffened shell structure;

[0010] (4) Calculate the equivalent density of each solid element in the stiffener design domain according to the obtained equivalent density of the thin shell control points, so as to obtain the final porous stiffener distribution on the thin shell structure and obtain the finally optimized porous stiffened shell structure.

[0011] Furthermore, use the Bézier extraction technology to decompose the high-order NURBS element mesh into a series of C 0 continuous Bézier elements, and the corresponding formula is:

[0012]

[0013] In the formula, P b and P are the Bézier control points and NURBS control points respectively; W and W b represent the weight diagonal matrices of the NURBS and Bézier elements respectively; R and B represent the NURBS basis function and the Bernstein basis function respectively; The expression of W is:

[0014] W(ξ,η)=wT CB(ξ, η)

[0015] Where w represents the NURBS weight vector and C is the coefficient matrix of the Bézier extraction operator.

[0016] Furthermore, the formula for the strain energy density per unit area of the isogeometric Kirchhoff–Love shell element is:

[0017]

[0018] Where ε and κ represent the membrane strain and bending strain tensors respectively, N and M are the force and moment stress resultant tensors respectively, D is the material tensor, and t is the thickness of the thin shell.

[0019] Furthermore, the formula for the element stiffness matrix of the Kirchhoff–Love thin shell is:

[0020]

[0021] Where J 1 and J 2 are the Jacobian matrices mapping from the parametric domain to the physical domain and from the parent domain to the parametric domain respectively; D e represents the element material tensor, Ω e is the element region; B m and B b represent the membrane and bending strain displacement matrices respectively;

[0022] The formula corresponding to the stiffness matrix of the solid-shell coupling element is:

[0023]

[0024] Where and represent the stiffnesses at the "coupling" and "solid" control points respectively, N c is the number of coupling control points, and N d is the total number of control points of the solid element.

[0025] Furthermore, the expression for the total stiffness matrix of the entire stiffened shell structure:

[0026]

[0027] Where represents the equivalent density of the e-th solid element in the l-th layer, and E min is a minimum value set to avoid singularity of the stiffness matrix; N e is the number of elements per layer, t r is the number of element layers; and respectively represent the stiffness matrices of the coupled elements and solid elements after equivalent density interpolation.

[0028] Furthermore, the expression of the displacement penalty term for the displacement continuity of the strong thin-shell structure is:

[0029]

[0030] where the superscripts A and B respectively represent two coupled thin-shell regions and α d represents the displacement penalty parameter of the two-dimensional NURBS patch, and respectively represent and the displacements at the coupling interface between them;

[0031] The expression of the rotation penalty term for enhancing the rotation continuity of the thin-shell structure is:

[0032]

[0033] where, and are respectively and the rotations at the coupling interface between them, α r represents the rotation penalty parameter of the two-dimensional NURBS patch, is the rotation along the normal direction of the coupling interface on is the in-plane unit normal vector of Γ shell at the coupling interface on

[0034] Furthermore, the expression of the penalty term for enhancing the displacement continuity of the stiffener structure is:

[0035]

[0036] where, α s represents the displacement penalty parameter of the three-dimensional NURBS patch; and respectively represent and the displacements at the coupling interface on solid is and the coupling interface between them.

[0037] Furthermore, the mathematical expression of the full-scale multi-patch isogeometric topology optimization model for the porous stiffened shell is:

[0038] Find: ρ i(i = 1, 2, …, n)

[0039]

[0040] In the formula, ρ i is the equivalent density of the control points of the thin shell, i.e., the design variable, J is the objective function with the minimum compliance as the goal, g V and g L respectively represent the global and local volume constraints, V max is the specified upper limit of the global volume fraction, ν is the volume of each element, L c represents the local volume fraction of the c-th control point, L max is the specified upper limit of the local volume fraction, u is the displacement vector, and f is the load vector.

[0041] Generally speaking, compared with the prior art through the above technical solutions conceived by the present invention, the full-scale design method of porous stiffened shells based on multi-patch isogeometric topology optimization provided by the present invention mainly has the following beneficial effects:

[0042] 1. By using Kirchhoff–Love shell elements with the same degrees of freedom as solid elements to describe the shell, the present invention can avoid stiffness loss caused by the mismatch of degrees of freedom between solid elements and shell elements, improve applicability, and can be applied to the stiffened design of thin shell structures with complex geometric shapes to obtain porous stiffened shells with high robustness.

[0043] 2. The present invention adopts local volume constraints with an adaptive allocation strategy, and automatically scales the influence radius of the local volume constraints at each control point according to the displacement field calculated in each iteration, and a bone-like porous stiffened shell structure with both high stiffness and high robustness can be obtained.

[0044] 3. By using solid elements instead of beam elements to describe the stiffeners, the present invention can flexibly construct the stiffened design domain on thin shell structures with complex geometric shapes, and can optimize the layout of stiffeners for thin shell structures with complex geometric shapes, with a wide range of applications.

[0045] 4. The full-scale design method of porous stiffened shells based on multi-patch isogeometric topology optimization provided by the present invention can efficiently and accurately optimize the layout of stiffeners for thin shell structures with complex geometric shapes, with a high degree of design freedom. BRIEF DESCRIPTION OF THE DRAWINGS

[0046] Figure 1 is a flowchart of a full-scale design method of porous stiffened shells based on multi-patch isogeometric topology optimization provided by the present invention;

[0047] Figure 2 is a schematic diagram of the Bézier extraction technology constructed by the present invention;

[0048] Figure 3 It is a schematic diagram of the solid-shell coupling unit constructed by the present invention;

[0049] Figure 4 It is a schematic diagram of the coupling interface between multiple NURBS patches of the stiffened shell constructed by the present invention;

[0050] Figure 5 It is a schematic diagram of the mapping constraint constructed by the present invention;

[0051] Figure 6 It is a schematic diagram of the structural design domain and boundary conditions of the stiffened shell constructed by the present invention;

[0052] Figure 7 It is a schematic diagram of the optimization iteration curve of the stiffened shell constructed by the present invention;

[0053] Figure 8 It is a distribution diagram of the stiffeners obtained by optimization constructed by the present invention;

[0054] Figure 9 It is a complete structure diagram of the stiffened shell obtained by optimization constructed by the present invention. Specific implementation manners

[0055] In order to make the objectives, technical solutions and advantages of the present invention clearer and more understandable, the present invention will be further described in detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are only used to explain the present invention and are not used to limit the present invention. In addition, the technical features involved in the various embodiments of the present invention described below can be combined with each other as long as they do not conflict with each other.

[0056] Please refer to Figure 1 , the present invention provides a full-scale design method for porous stiffened shells based on multi-patch isogeometric topology optimization. The method constructs an isogeometric Kirchhoff–Love thin shell model using an isogeometric analysis method based on Bézier extraction to perform efficient and high-precision performance analysis on the thin shell structure. At the same time, the penalty method is used to ensure the displacement and rotation continuity between multiple NURBS patches. The layout of the stiffeners on the thin shell is represented by setting an equivalent density field on the control grid, and mapping constraints are introduced to avoid obtaining a stiffener structure with overhang features. Local volume constraints are imposed on each equivalent density to obtain the distribution of bone-shaped porous stiffeners. An adaptive allocation strategy based on the displacement field is adopted for local volume constraints to reduce the influence of local volume constraints on the stiffness performance of the stiffener structure. Finally, a full-scale multi-patch isogeometric topology optimization model is constructed with the minimum compliance as the objective function and the total volume fraction and local volume fraction as the constraint functions to optimize the distribution of the stiffeners on the thin shell, obtaining a porous stiffened shell with high stiffness and high damage robustness and realizing topology optimization.

[0057] The method mainly includes the following steps:

[0058] Step 1: First, construct the geometric model of the thin-shell structure and the geometric model of the stiffener region using multiple two-dimensional NURBS patches and multiple three-dimensional NURBS patches respectively, and divide the thin-shell structure and the stiffener region into high-order NURBS element meshes respectively.

[0059] Dividing the thin-shell structure and the stiffener region into high-order NURBS element meshes ensures that the meshes of the thin-shell structure and the stiffener region are completely matched.

[0060] Step 2: Use isogeometric Kirchhoff–Love shell elements to describe the thin-shell structure, use three-dimensional solid elements to describe the stiffener region, introduce the Bézier extraction technology to decompose the high-order NURBS element mesh into a series of C 0 continuous Bézier elements, and then add each element in the shell element stiffness matrix to the element at the corresponding degree of freedom in the corresponding solid element stiffness matrix to construct a solid-shell coupled element. At the same time, introduce displacement penalty terms and rotation penalty terms between multiple NURBS patches respectively.

[0061] Among them, adding each element in the shell element stiffness matrix to the element at the corresponding degree of freedom in the corresponding solid element stiffness matrix constructs a solid-shell coupled element to realize the integrated performance analysis of the stiffener and the thin shell; at the same time, introduce displacement penalty terms and rotation penalty terms between multiple NURBS patches respectively to ensure the displacement and rotation continuity between multiple NURBS patches.

[0062] Using the Bézier extraction technology to decompose the high-order NURBS element mesh into a series of C 0 continuous Bézier elements, and the corresponding formula is:

[0063]

[0064] In the formula, P b and P are the Bézier control points and the NURBS control points respectively; W and W b represent the weight diagonal matrices of the NURBS and Bézier elements respectively; R and B represent the NURBS basis function and the Bernstein basis function respectively; the expression of W is:

[0065] W(ξ,η) = w T CB(ξ,η)

[0066] Among them, w represents the NURBS weight vector, and C is the coefficient matrix of the Bézier extraction operator.

[0067] The formula for the strain energy density per unit area obtained from the isogeometric Kirchhoff–Love shell element is as follows:

[0068]

[0069] where ε and κ represent the membrane strain and bending strain tensors respectively, N and M are the force and moment stress resultant tensors respectively, D is the material tensor, and t is the thickness of the thin shell.

[0070] Using isogeometric analysis based on Bézier extraction to discretize the Kirchhoff–Love thin shell, the corresponding formula is:

[0071]

[0072] where r and u represent the position vector and displacement vector respectively.

[0073] The formula for the element stiffness matrix of the Kirchhoff–Love thin shell is:

[0074]

[0075] where J 1 and J 2 are the Jacobian matrices mapping from the parametric domain to the physical domain and from the parent domain to the parametric domain respectively; D e represents the element material tensor, Ω e is the element area; B m and B b represent the membrane and bending strain displacement matrices respectively.

[0076] The formula for the stiffness matrix of the solid-shell coupling element is:

[0077]

[0078] where and represent the stiffnesses at the "coupling" and "solid" control points respectively, N c is the number of coupling control points, and N d is the total number of control points of the solid element.

[0079] The expression for the total stiffness matrix of the entire stiffened shell structure:

[0080]

[0081] where represents the equivalent density of the e-th solid element in the l-th layer, and E min is a minimum value set to avoid singularity of the stiffness matrix. N e is the number of elements per layer, and t ris the number of cell layers. and respectively represent the stiffness matrices of the coupled cells and solid cells after equivalent density interpolation.

[0082] The expression of the displacement penalty term for the displacement continuity of the strong thin-shell structure is:

[0083]

[0084] where the superscripts A and B respectively represent two coupled thin-shell regions and α d represents the displacement penalty parameter of the 2D NURBS patch, and respectively represent and the displacements at the coupling interface between them.

[0085] The expression of the rotation penalty term for enhancing the rotation continuity of the thin-shell structure is:

[0086]

[0087] where, and are respectively and the rotations at the coupling interface between them, α r represents the rotation penalty parameter of the 2D NURBS patch, is the rotation along the normal direction of the coupling interface on is the in-plane unit normal vector of Γ shell at the coupling interface on

[0088] The expression of the penalty term for enhancing the displacement continuity of the stiffener structure is:

[0089]

[0090] where, α s represents the displacement penalty parameter of the 3D NURBS patch. and respectively represent and the displacements at the coupling interface on Γ solid is and the coupling interface between them.

[0091] Step 3: Set the equivalent density on each solid element as the design variable. By applying mapping constraints, transform this design variable into the equivalent density on the thin-shell control points. At the same time, introduce local volume constraints on the design variable to obtain bone-like porous stiffeners, and then construct a full-scale multi-patch isogeometric topology optimization model for the porous stiffened shell. The full-scale multi-patch isogeometric topology optimization model is optimized iteratively to obtain an optimized porous stiffened shell structure.

[0092] Among them, by applying mapping constraints, transform this design variable into the equivalent density on the thin-shell control points to avoid the generation of stiffener structures with overhang features;

[0093] The mathematical expression of the full-scale multi-patch isogeometric topology optimization model for the porous stiffened shell is:

[0094] Find: ρ i (i = 1, 2, …, n)

[0095]

[0096] In the formula, ρ i is the equivalent density of the control points of the thin shell, that is, the design variable. J is the objective function with the minimum compliance as the goal. g V and g L respectively represent the global and local volume constraints. V max is the specified upper limit of the global volume fraction. ν is the volume of each element. L c represents the local volume fraction of the c-th control point. L max is the specified upper limit of the local volume fraction. u is the displacement vector. f is the load vector. The expression of the local volume fraction L c is as follows:

[0097]

[0098] Among them, n c is the set of adjacent control points within the influence radius r L of the i-th control point.

[0099] Introduce mapping constraints to ensure that the stiffeners are evenly distributed in the thickness direction. The corresponding expression is as follows:

[0100]

[0101] Among them, is the equivalent density of the e-th shell element, which is used to force the solid elements with the same in-plane position in different layers to have the same equivalent density. The equivalent density field established by the equivalent density ρ i of the control points on the shell mesh and the corresponding NURBS basis function R i is The equivalent density of each shell element can be obtained Equivalent density field The expression is as follows:

[0102]

[0103] where n is the number of control points of the thin shell; τ represents the penalty factor, and φ i (ρ i ) is the Heaviside step function.

[0104] In this embodiment, a local volume constraint adaptive allocation strategy based on the displacement field is proposed to avoid significant loss of the stiffness performance of the optimized stiffened shell. The expression of the mapping relationship between the influence radius of the control point and the displacement is as follows:

[0105]

[0106] where r L represents the influence radius field, and are the specified upper and lower limits of the influence radius respectively. U is the displacement field, and U max and U min are the maximum and minimum values of the displacement respectively.

[0107] In this embodiment, the method for updating the design variables is the moving asymptote method based on sensitivity information.

[0108] Step 4: Calculate the equivalent density of each solid element in the stiffener design domain according to the obtained equivalent density of the optimized control points of the thin shell, so as to obtain the final porous stiffener distribution on the thin shell structure, and obtain the finally optimized porous stiffened shell structure.

[0109] The following uses specific embodiments to further elaborate on the present invention in detail.

[0110] Embodiment 1

[0111] In this embodiment, the design domain, load and boundary conditions of the stiffened shell to be optimized are as Figure 4As shown, it is a hemispherical stiffened shell with L = 200 mm, H = 86.6 mm, r = 50 mm, and H = 6 mm. Among them, a uniformly distributed load F = 10000 N is applied to the top of the hemispherical stiffened shell, and the bottom of the hemispherical stiffened shell is fixed. Here, stainless steel is selected as the structural material, with a Young's modulus of 210 GPa and a Poisson's ratio of 0.3. The thin shell area of the hemispherical shell consists of four two-dimensional NURBS patches, and the stiffener design domain consists of four three-dimensional NURBS patches. Each three-dimensional NURBS patch is divided into 100×100×2 Bezier elements. The overall volume constraint is set to 0.4, and the local volume constraint is set to 0.6. In order to reduce the number of iteration steps, the initial design variables are uniformly set to 0.5.

[0112] The full-scale design method of a porous stiffened shell based on multi-patch isogeometric topology optimization provided in Embodiment 1 of the present invention includes the following steps:

[0113] Step 1, first construct the geometric model of the thin shell structure and the geometric model of the stiffener region with multiple two-dimensional NURBS patches and multiple three-dimensional NURBS patches respectively, and divide the thin shell structure and the stiffener region into high-order NURBS element meshes respectively.

[0114] An isogeometric Kirchhoff–Love thin shell model is constructed using an isogeometric analysis method based on Bézier extraction to perform efficient and high-precision performance analysis on the thin shell structure.

[0115] Specifically, it includes the following sub-steps:

[0116] (1.1) Use the Bézier extraction technique to convert the NURBS elements divided by isogeometric analysis into Bézier elements. The specific formula is:

[0117]

[0118] Among them, P b and P are the Bézier control points and NURBS control points respectively; W and W b represent the weight diagonal matrices of the NURBS and Bézier elements respectively; R and B represent the NURBS basis function and the Bernstein basis function respectively. The calculation formula of W is as follows:

[0119] W(ξ,η) = w T CB(ξ,η)

[0120] Among them, w represents the NURBS weight vector, and C is the coefficient matrix of the Bézier extraction operator.

[0121] (1.2) Construct the Kirchhoff–Love thin shell model. The formula for the strain energy density per unit area of the Kirchhoff–Love shell is:

[0122]

[0123] where ε and κ represent the membrane strain and bending strain tensors respectively, N and M are the force and moment stress resultant tensors respectively, D is the material tensor, and t is the thickness of the thin shell.

[0124] (1.3) Discretize the Kirchhoff–Love thin shell model using isogeometric analysis based on Bézier extraction. The specific formula is:

[0125]

[0126] where r and u represent the position vector and displacement vector respectively.

[0127] The formula for the element stiffness matrix in the constructed isogeometric Kirchhoff–Love thin shell model is as follows:

[0128]

[0129] where J 1 and J 2 are the Jacobian matrices mapping from the parametric domain to the physical domain and from the parent domain to the parametric domain respectively; D e represents the element material tensor, Ω e is the element region; B m and B b represent the membrane and bending strain displacement matrices respectively.

[0130] Step 2: Use shell elements to describe the thin shell region and solid elements to describe the stiffener region. Introduce the Bézier extraction technique to decompose the high-order NURBS element mesh into a series of Bézier elements with the same data structure and continuous C 0 Then add each element in the shell element stiffness matrix to the element at the corresponding degree of freedom in the solid element stiffness matrix to construct the solid-shell coupled element. At the same time, use the penalty method to ensure the displacement and rotation continuity between multiple NURBS patches.

[0131] Specifically, it includes the following sub-steps:

[0132] (2.1) Ensure that the shell mesh and solid mesh are completely matched when meshing the thin shell region and the stiffener region. Add each element in each shell element stiffness matrix to the element at the corresponding degree of freedom in the adjacent solid element stiffness matrix to construct the stiffness matrix of the solid-shell coupled element. The specific formula is:

[0133]

[0134] Among them, and represent the stiffness at the "coupling" and "entity" control points respectively. N c is the number of coupling control points, and N d is the total number of control points of the entity elements.

[0135] For the discretization of the stiffener region by multi-layer entity elements, the total stiffness matrix of the entire stiffened shell structure is:

[0136]

[0137] Among them, represents the equivalent density of the e-th entity element in the l-th layer, and E min is a minimum value set to avoid singularity of the stiffness matrix. N e is the number of elements per layer, and t r is the number of element layers. and represent the stiffness matrices of the coupled elements and entity elements interpolated through the equivalent density respectively.

[0138] (2.2) Introduce a penalty term to the thin shell structure to enhance the displacement and rotation continuity between multiple two-dimensional NURBS patches describing the thin shell structure; the specific formula of the penalty term to ensure displacement continuity is:

[0139]

[0140] Among them, the superscripts A and B represent two coupled thin shell regions respectively and α d represents the displacement penalty parameter of the two-dimensional NURBS patch. and represent respectively and the displacements at the coupling interface between them.

[0141] The specific formula of the penalty term to ensure rotation continuity is:

[0142]

[0143] Among them, and are respectively and the rotations at the coupling interface between them, and α r represents the rotation penalty parameter of the two-dimensional NURBS patch, is the rotation along the normal direction of the coupling interface on is Γ at the upper coupling interface shell in-plane unit normal vector of the surface.

[0144] (2.3) Introduce a penalty term for the stiffener structure to enhance the displacement continuity between multiple 3D NURBS patches describing the stiffener structure. The specific formula for the penalty term is as follows:

[0145]

[0146] where α s represents the displacement penalty parameter of the 3D NURBS patch. and respectively represent and the displacements at the upper coupling interface. Γ solid is and the coupling interface between.

[0147] Step 3: Introduce mapping constraints to avoid the generation of stiffener structures with overhang features, apply local volume constraints to obtain bone-like porous stiffeners, construct a full-scale multi-patch isogeometric topology optimization model for porous stiffened shells, and use the moving asymptote method to optimize the stiffener distribution on the thin shell to obtain a porous stiffened shell structure with both high stiffness and high robustness.

[0148] Specifically, it includes the following sub-steps:

[0149] (3.1) The mathematical expression of the full-scale multi-patch isogeometric topology optimization model for porous stiffened shells is:

[0150] Find: ρ i (i = 1, 2, …, n)

[0151]

[0152] where ρ i is the equivalent density of the control points of the thin shell, i.e., the design variable, J is the objective function with the minimum compliance as the goal, g V and g L respectively represent the global and local volume constraints, V max is the specified upper limit of the global volume fraction, ν is the volume of each element, L c represents the local volume fraction of the c-th control point, L max is the specified upper limit of the local volume fraction, u is the displacement vector, and f is the load vector.

[0153] Specifically, introduce mapping constraints to ensure the uniform distribution of the stiffeners in the thickness direction, and its form is as follows:

[0154]

[0155] Among them, is the equivalent density of the e-th shell element, which is used to force the solid elements with the same in-plane position in different layers to have the same equivalent density. The equivalent density ρ at the control points on the shell mesh i and the corresponding NURBS basis function R i are used to establish the equivalent density field from which the equivalent density of each shell element can be obtained Equivalent density field The formula is as follows:

[0156]

[0157] Among them, n is the number of control points of the thin shell. τ represents the penalty factor. φ i (ρ i ) is the Heaviside step function, which is used to promote the optimization result to tend to the 0-1 solution. The formula is as follows:

[0158]

[0159] Among them, σ represents the projection parameter. To maintain the stability of the iterative process, the initial value σ = 1 is specified, and then its value is doubled every 50 iterations until σ = 32.

[0160] Specifically, the local volume fraction L c is calculated as follows:

[0161]

[0162] Among them, n c is the set of adjacent control points within the influence radius r L of the i-th control point.

[0163] To avoid significant loss of the stiffness performance of the stiffened shell after optimization, an adaptive allocation strategy of local volume constraint based on the displacement field is adopted. The mapping relationship between the influence radius of the control point and the displacement is:

[0164]

[0165] Among them, r L represents the influence radius field, and are the specified upper and lower limits of the influence radius, taking 20 and 6 respectively. U is the displacement field, U max and U minThey are the maximum and minimum values of displacement respectively. During the optimization process, the displacement field will change continuously with the update of design variables, and the influence radius field will adaptively change with the displacement field calculated in each iteration.

[0166] (3.2) Calculate the sensitivities of the objective function and the global and local volume constraint functions with respect to the design variables. The sensitivity expression of the objective function with respect to the design variables is as follows:

[0167]

[0168] The sensitivity calculation formula of the global volume constraint condition with respect to the design variables is:

[0169]

[0170] The sensitivity calculation formula of the local volume constraint condition with respect to the design variables is:

[0171]

[0172] When updating the design variables, filtering is carried out by using the average value of the adjacent control point densities to replace the current control point density, so as to avoid numerical instability phenomena such as checkerboards and mesh dependencies. Here, the filtering radius is taken as 2.

[0173] (3.3) Substitute the sensitivities of the objective function and the global and local volume constraint functions with respect to the design variables obtained in step (3.2) into the moving asymptote method to update the design variables.

[0174] (3.4) Construct the convergence condition according to the maximum change value of the design variables between two iterations. If the maximum change value of the design variables between two iterations is less than 1% or the number of iteration steps reaches 600 steps, stop the iteration and output the finally updated equivalent density ρ i , if the convergence condition is not satisfied, return to step (3.1) to continue updating the design variables.

[0175] Step four, according to the equivalent density ρ i output after satisfying the convergence condition in step three, obtain the equivalent density of each solid element in the stiffener design domain According to obtain the final porous stiffener distribution on the thin shell, and get the final porous stiffener, thus realizing the topology optimization process.

[0176] Please refer to Figures 2 to 9 , the following takes the design of a hemispherical stiffened shell as an example to further illustrate the present invention.

[0177] As Figure 2 shown is the schematic diagram of the Bézier extraction technique.

[0178] AsFigure 3 Shown is a schematic diagram of a solid-shell coupling element, where the shell element is closely attached to the bottom surface of the solid element and the control point meshes of the two are completely matched.

[0179] As Figure 4 Shown is a schematic diagram of the coupling interface between multiple NURBS patches of a stiffened shell, where and represent two adjacent 2D NURBS patches used to describe the thin shell structure, and represent two adjacent 3D NURBS patches used to describe the stiffener structure.

[0180] As Figure 5 Shown is a schematic diagram of mapping constraints, where by forcing the equivalent densities of solid elements with the same in-plane position in different layers to be uniformly equal to the equivalent density of the shell element at the corresponding position, the optimization of stiffeners with voids along the thickness direction is avoided.

[0181] As Figure 6 Shown is the design domain of a hemispherical stiffened shell with L = 200 mm, H = 86.6 mm, r = 50 mm, and H = 6 mm, where a uniformly distributed load F = 10,000 N is applied to the top of the hemispherical stiffened shell, and the bottom of the hemispherical stiffened shell is fixed. Stainless steel is selected as the structural material, with a Young's modulus of 210 GPa and a Poisson's ratio of 0.3. The thin shell region of the hemispherical shell consists of four 2D NURBS patches, and the stiffener design domain consists of four 3D NURBS patches. Each 3D NURBS patch is divided into 100×100×2 Bézier elements; the overall volume constraint is set to 0.4, and the local volume constraint is set to 0.6. To reduce the number of iteration steps, the initial design variables are uniformly set to 0.5.

[0182] As Figure 7 Shown is the optimization iteration curve of the hemispherical stiffened shell. The iteration process is relatively stable, and while the compliance value decreases, the volume fraction remains unchanged at the upper limit of the constraint. As Figure 8 Shown is the distribution of stiffeners on the optimized hemispherical thin shell. As Figure 9 Shown is the complete structural diagram of the optimized hemispherical stiffened shell. It can be seen from the figure that compared with the traditional method, the full-scale design method of porous stiffened shells based on multi-patch isogeometric topology optimization provided by the present invention can optimize a porous stiffened shell structure with both high stiffness and high damage robustness for complex thin shell structures, with high analysis and optimization accuracy, high design freedom, and wide application range.

[0183] Those skilled in the art can easily understand that the above are only the preferred embodiments of the present invention and are not intended to limit the present invention. Any modifications, equivalent replacements, and improvements made within the spirit and principles of the present invention should be included in the protection scope of the present invention.

Claims

1. A full-scale design method for porous reinforced shells based on multi-piece isogeometric topology optimization, characterized in that: The method comprises the following steps: (1) First, a geometric model of the thin shell structure and a geometric model of the stiffener region are constructed using multiple two-dimensional NURBS patches and multiple three-dimensional NURBS patches, and the thin shell structure and the stiffener region are divided into high-order NURBS unit grids; (2) The isogeometric Kirchhoff–Love shell element is used to describe the thin shell structure, the three-dimensional solid element is used to describe the stiffener area, and the Bézier extraction technology is introduced to decompose the high-order NURBS unit grid into a series of C 0 Continuous Bézier unit, then each element in the shell unit stiffness matrix is ​​added to the element at the corresponding degree of freedom in the corresponding solid unit stiffness matrix to construct a solid-shell coupling unit, and at the same time, displacement penalty terms and rotation penalty terms are introduced between multiple NURBS patches; (3) setting an equivalent density as a design variable on each solid unit, converting the design variable into an equivalent density on a thin shell control point by applying a mapping constraint, and introducing a local volume constraint on the design variable to obtain bone-like porous reinforcement ribs, thereby constructing a full-scale multi-piece isogeometric topology optimization model for porous reinforced shells. The full-scale multi-piece isogeometric topology optimization model is optimized and iterated to obtain an optimized porous reinforced shell structure; (4) The equivalent density of each solid unit in the stiffener design domain is calculated based on the obtained optimized equivalent density of the thin shell control points, so as to obtain the final porous stiffener distribution on the thin shell structure and obtain the final optimized porous stiffened shell structure.

2. The full-scale design method of porous reinforced shell based on multi-piece equal geometric topology optimization according to claim 1, characterized in that: The Bézier extraction technique is used to decompose the high-order NURBS unit grid into a series of C 0 The corresponding formula for continuous Bézier units is: Where P b and P are Bézier control points and NURBS control points respectively; W and W b Represent the weight diagonal matrices of NURBS and Bézier elements respectively; R and B represent NURBS basis functions and Bernstein basis functions respectively; the expression of W is: W(ξ,η)=w T CB(ξ,η) Where w represents the NURBS weight vector and C is the coefficient matrix of the Bézier extraction operator.

3. The full-scale design method of porous reinforced shell based on multi-piece equal geometric topology optimization according to claim 2, characterized in that: The formula for the strain energy density per unit area of ​​the isogeometric Kirchhoff–Love shell element is: where ε and κ represent the membrane strain and bending strain tensors, respectively, N and M are the force and moment stress resultant tensors, respectively, D is the material tensor, and t is the thickness of the shell.

4. The full-scale design method of porous reinforced shell based on multi-piece equal geometric topology optimization according to claim 1, characterized in that: The element stiffness matrix formula for the Kirchhoff–Love thin shell is: Where J1 and J2 are the Jacobian matrices mapping from the parameter domain to the physical domain and from the parent domain to the parameter domain, respectively; D e Represents the element material tensor, Ω e is the unit area; B m and B b denote the membrane and bending strain displacement matrices, respectively; The formula corresponding to the stiffness matrix of the solid-shell coupling element is: in, and Represent the stiffness of the coupling and solid control points, N c is the number of coupling control points, N d is the total number of control points of the solid element.

5. The full-scale design method of porous reinforced shell based on multi-piece equal geometric topology optimization according to claim 4, characterized in that: The expression of the total stiffness matrix of the entire stiffened shell structure is: in, represents the equivalent density of the e-th entity unit in the l-th layer, E min To avoid the singularity of the stiffness matrix, set the minimum value; N e is the number of units per layer, t r is the number of unit layers; and Respectively represent the equivalent density Interpolated stiffness matrices of coupled and solid elements.

6. The full-scale design method of porous reinforced shell based on multi-piece equal geometric topology optimization according to claim 1, characterized in that: The expression of the displacement penalty term of the displacement continuity of the strong thin shell structure is: The superscripts A and B represent two coupled thin shell regions. and α d Represents the displacement penalty parameter of a two-dimensional NURBS patch, and Respectively and The displacement at the coupling interface between them; The expression of the rotation penalty term that enhances the rotational continuity of the thin shell structure is: in, and They are and The rotation at the coupling interface between r Represents the rotation penalty parameter of a two-dimensional NURBS patch, is along The normal direction of the upper coupling interface rotates, yes Γ at the upper coupling interface shell The unit normal vector in the plane of .

7. The full-scale design method of porous reinforced shell based on multi-piece equal geometric topology optimization according to claim 6, characterized in that: The expression of the penalty term for enhancing the displacement continuity of the stiffener structure is: Among them, α s Represents the displacement penalty parameter of the three-dimensional NURBS patch; and Respectively and Displacement at the upper coupling interface; Γ solid yes and The coupling interface between them.

8. The full-scale design method of porous reinforced shell based on multi-piece equal geometric topology optimization according to claim 7, characterized in that: The mathematical expression of the full-scale multi-piece isogeometric topology optimization model for porous reinforced shells is: Find:ρ i (i=1,2,…,n) In the formula, ρ i is the equivalent density of the control points of the thin shell, i.e., the design variable, J is the objective function with the goal of minimizing flexibility, and g V and g L Represent the global and local volume constraints, V max is the specified upper limit on the global volume fraction, ν is the volume of each element, and L c represents the local volume fraction of the cth control point, L max is the specified upper limit of the local volume fraction, u is the displacement vector, and f is the load vector.

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