CN112926161B - Optimization method for spatial curved shell structure shape - Google Patents

Optimization method for spatial curved shell structure shape Download PDF

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CN112926161B
CN112926161B CN202110349665.3A CN202110349665A CN112926161B CN 112926161 B CN112926161 B CN 112926161B CN 202110349665 A CN202110349665 A CN 202110349665A CN 112926161 B CN112926161 B CN 112926161B
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李霞
周克民
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Huaqiao University
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Abstract

The invention discloses a method for optimizing the shape of a space curved shell structure, and aims to find a method for designing the space curved shell structure with the minimum stress constrained volume under the action of vertical load, which is used for conceptual optimization design of the outline of the space curved shell. This method assumes that the anisotropic material is not uniformly distributed within the spatial design domain. The vertical load is distributed in the material in the form of its own weight according to the density of the material. And analyzing the structure by adopting a finite element method to obtain a stress and strain distribution field. The density and direction of the material at the node position are used as optimization design variables. And optimizing the density and the direction of the material at the node position by adopting a stress ratio formula of a full stress criterion according to the main stress direction of the node position and the strain magnitude of the main stress direction. And interpolating by a shape function to obtain the density and the direction of the material at any position. And structural analysis is carried out again. And repeating the iteration until convergence, and realizing the optimal distribution of the material in the design domain. The resulting material is distributed within a thin curved surface to obtain the shape and thickness of the curved surface. The method does not need initial structure and design experience, and can directly optimize and form the curved surface.

Description

Optimization method for spatial curved shell structure shape
Technical Field
The invention relates to the field of materials, in particular to a method for optimizing the shape of a space curved shell structure.
Background
The space curved shell structure has the advantages of reasonable stress, material saving, small occupied space, attractive appearance and the like, and is frequently used in the fields of aerospace, materials, machinery, civil engineering, ships, water conservancy and the like. These space curved shell structures are not only required to meet the functional requirements of use, but also are desired to use as little material as possible. How to design the size and shape of the space curved shell structure of various materials under different load and supporting conditions is a general concern in the structural design.
Some design methods have been developed for reticulated shell structures, cable-mesh structures, and tensioned membrane structures. However, these design methods require an initial design (or conceptual design) solution. And the initial design solution relies primarily on the experience of the designer. At present, some structure Optimization design methods, such as a Homogenization Method (Homogenization Method), an evolution structure Optimization Method (ESO), and an Isotropic Solid penalty intermediate density Method (SIMP), mainly aim at researching a plane problem, also aim at a small amount of research and design space problems, but are also general space problems, and are not suitable for designing a space curved shell.
The space curved shell structure topological optimization concept design needs to solve several problems: (a) A construction method of a space orthogonal anisotropic material model; (b) Forming a mathematical expression of a spatial orthotropic material stiffness matrix of finite element calculation; (c) A description method of a continuous distribution field of a non-uniform space orthotropic material; and (d) a topological optimization iterative method of the space curved shell structure.
Disclosure of Invention
The invention mainly aims to overcome the defects that the conventional structure optimization design method is mainly used for researching the plane problem and is not suitable for the design of a space curved shell, and provides a space curved shell structure shape optimization method aiming at the conceptual design problem of the problem of minimum single-working-condition stress constraint volume of the space curved shell structure.
The invention adopts the following technical scheme:
a method for optimizing the shape of a spatial curved shell structure is characterized by comprising the following steps:
1) Setting a design domain, dividing the design domain by using a limited unit, and initializing the direction and density of a material at a node;
2) Distributing the load to each node in the design domain according to the material density distribution;
3) Establishing a structural rigidity matrix of a finite unit, assembling into a rigidity equation, introducing displacement constraint, solving the rigidity equation to obtain a node displacement vector, wherein the structural rigidity matrix is as follows:
Figure GDA0003866574780000021
in the formula, S e Representing a set of nodes belonging to a finite element e, t bj Which represents the density of the material and is,
Figure GDA0003866574780000022
e is the modulus of elasticity, V e Representing the range of each finite elementB is the set matrix, T denotes transpose, A r Is a constant matrix independent of node location, r =1,2.. 6,g r (n bj ) About the material direction
Figure GDA0003866574780000023
As a function of (a) or (b),
Figure GDA0003866574780000024
and
Figure GDA0003866574780000025
representing the components of the material direction vector on each coordinate axis; when a regular finite element is used, the matrix is a constant matrix;
the stiffness equation is as follows:
KU＝F
wherein F is a load column vector, and U is a node displacement column vector;
principal stress is determined by solving the eigenvector m of the equation bj And the eigenvalue σ bj Obtaining:
Figure GDA0003866574780000026
the element of the 1 st matrix in the formula is a stress component, and is calculated by the following formula
[σ x σ y σ z τ yz τ zy τ xy ] T ＝DBU e
D is an elastic matrix, B is an aggregate matrix, U e Is a node displacement column vector of finite elements;
solving the stiffness equation to obtain a node displacement column vector U,
the resulting structure volume was as follows:
Figure GDA0003866574780000027
in the formula N j Is a shape function of a finite element;
4) Calculating the main stress of the node position according to the node displacement vector, aligning the main direction of the material at the node position with the main stress direction, and optimizing the direction and the density of the material according to the full stress criterion;
5) Judging whether the maximum values of the relative variation and the direction absolute variation of the material density of each node position in the two continuous iteration processes are less than or equal to a preset value, returning to the step 2 if not, carrying out structural analysis and optimization again, and entering the step 6 if so;
6) And calculating the thickness of the material and the position of the middle surface of the curved shell structure to form an optimized middle surface of the spatial curved shell structure.
In the step 1), an eight-node cuboid unit is adopted to subdivide the design domain, and all edges of the eight-node cuboid unit keep in the vertical direction to form a plurality of vertical lines.
The direction and density of the initialization material at the nodes are specifically as follows: three main direction initial values of the three orthogonal materials at any node j position of the material are set as coordinate axis directions:
Figure GDA0003866574780000031
the initial density of the material in the three material main axis directions is t bj ＝0.2,b＝1,2,3。
In the step 2), the vertical load distributed along the vertical direction is distributed to the edges of the finite elements according to the virtual work principle, and the expression is as follows:
Figure GDA0003866574780000032
in the form of an integral field V e Representing the extent of each finite element, N p (x, y, z) is a shape function, q (x, y) is a distribution function of vertical load in the horizontal direction, and x, y, z represents a spatial coordinate position.
In step 4), optimizing the density of the material at the position of the joint j according to a full stress criterion method
Figure GDA0003866574780000033
In the formula the superscript i denotes the iteration index,
Figure GDA0003866574780000034
is the principal stress, σ p Is the material allowable stress; in order to avoid singular of the structural rigidity matrix, the material density is limited
Figure GDA0003866574780000035
Figure GDA0003866574780000036
In step 4), the main direction of the material at the joint position is aligned with the main stress direction, which is expressed as follows
Figure GDA0003866574780000037
In the formula, m bj Indicating the principal stress direction.
In step 5), whether the maximum values of the relative variation and the direction absolute variation of the material density at each node position are small enough in the two successive iteration processes is checked, and the specific expression is as follows:
Figure GDA0003866574780000038
wherein δ is a preset value, | ∞is ∞ -norm.
The material thickness is calculated from the following formula,
Figure GDA0003866574780000039
in the formula1 st one
Figure GDA0003866574780000041
All units in the vertical direction representing the same horizontal position are accumulated; t is t b Is the density of the material at any position (x, y, z),
Figure GDA0003866574780000042
the position z of the surface in the curved shell structure c (x j ,y j ) Calculated from the centroid formula:
Figure GDA0003866574780000043
thereby forming an optimized curved shell structure midplane.
As can be seen from the above description of the present invention, compared with the prior art, the present invention has the following advantages:
according to the method, the vertical load is distributed on the material in a self-weight mode according to the density of the material, a finite element method is adopted to analyze the structure to obtain a stress and strain distribution field, and the density and the direction of the material at the node position are used as optimization design variables. And optimizing the density and direction of the material at the node position by adopting a stress ratio formula of a full stress criterion according to the main stress direction of the node position and the strain magnitude of the main stress direction, interpolating by a shape function to obtain the density and direction of the material at any position, and analyzing the structure again. And repeating the iteration until convergence, and realizing the optimal distribution of the material in the design domain. The resulting material is distributed within a thin curved surface to obtain the shape and thickness of the curved surface. The curved surface can be directly optimized and formed without initial structure and design experience
Drawings
FIG. 1 is a flow chart of the optimization method of the present invention.
FIG. 2 is an embodiment of a cube initial design domain and constraints.
FIG. 3 is an isometric view of the in-plane location and material density and orientation in the curved shell.
Fig. 4 is a section line of the curved shell in different sections parallel to the coordinate plane.
Fig. 5 illustrates the location of the sections in fig. 4.
The invention is described in further detail below with reference to the following figures and specific examples.
Detailed Description
The invention is further described below by means of specific embodiments.
Referring to fig. 1, a method for optimizing the shape of a spatial curved shell structure comprises the following steps:
1) A design domain is set and is split by finite elements, and the direction and density of the material at the nodes are initialized.
In the step, a design domain is firstly set according to the requirements of practical problems, and if the practical problems do not have any limitation on the area, the design domain which is large enough is selected as far as possible so as to obtain a better design result. In the invention, the eight-node cuboid unit can be adopted to subdivide the design domain, and other space units can be adopted for the irregular design domain. In order to facilitate the vertical load treatment, all the edges of the eight-node cuboid unit are kept in the vertical direction to form a plurality of vertical lines L p 。
The direction and density of the initialization material at the junction are specifically as follows: three-phase orthogonal isotropic materials are continuously and uniformly arranged in a design space, and three main direction initial values of the orthogonal materials at any node j position of the materials can be set to coordinate axis directions:
Figure GDA0003866574780000051
the initial density of the material in the three material main axis directions is t bj ＝0.2,b＝1,2,3。
2) The load is distributed to the nodes in the design domain according to the material density distribution.
In the step, vertical loads distributed along the vertical direction are distributed to the edges of the finite elements according to the virtual work principle, and the expression is as follows:
Figure GDA0003866574780000052
in the form of an integral field V e Representing the range of each finite element, N p (x, y, z) is a shape function, q (x, y) is a distribution function of vertical load in the horizontal direction, and x, y, z represent spatial coordinate positions.
3) And establishing a structural rigidity matrix of the finite element, assembling the structural rigidity matrix into a rigidity equation, introducing displacement constraint, and solving the rigidity equation to obtain a node displacement vector. In the step, a rigidity equation of the non-uniform anisotropic material is established, and displacement constraints such as fixed, hinged or movable hinged are introduced according to actual use conditions.
The elastic matrix of the material at the position of the node j is set as
Figure GDA0003866574780000053
In the formula g r (n bj ) About the direction of the material
Figure GDA0003866574780000054
As a function of (a) or (b),
Figure GDA0003866574780000055
A r (r =1,2.. 6) is a constant matrix independent of node location.
Figure GDA0003866574780000061
Figure GDA0003866574780000062
The elastic matrix at any position in the cell is
Figure GDA0003866574780000063
The structural stiffness matrix is then:
Figure GDA0003866574780000064
in the formula, S e Representing a set of nodes belonging to a finite element e, t bj Which represents the density of the material and is,
Figure GDA0003866574780000065
e is the modulus of elasticity, V e Representing the extent of each finite element, B is a geometric matrix, T represents a transpose, A r Is a constant matrix independent of node location, r =1,2.. 6,g r (n bj ) About the material direction
Figure GDA0003866574780000066
As a function of (a) or (b),
Figure GDA0003866574780000067
and
Figure GDA0003866574780000068
representing the component of the material direction vector on each coordinate axis; when a regular finite element is used, the matrix is a constant matrix;
and distributing vertical loads to the loads at each horizontal position along the vertical direction according to the material density to obtain a load column vector F.
Figure GDA0003866574780000069
The stiffness equation is established as follows:
KU＝F
and F is a load column vector, U is a node displacement column vector, a displacement constraint condition is introduced, and a stiffness equation is solved to obtain the node displacement column vector U.
The structure volumes obtained were further as follows:
Figure GDA00038665747800000610
in the formula N j Is a shape function of the finite element.
Principal stress is determined by solving the eigenvector m of the equation bj And the eigenvalue σ bj Obtaining:
Figure GDA0003866574780000071
the element of the 1 st matrix in the formula is a stress component, and is calculated by the following formula
[σ x σ y σ z τ yz τ zy τ xy ] T ＝DBU e
D is an elastic matrix, B is an aggregate matrix, U e Is a node-shift column vector of finite elements.
4) And calculating the main stress of the node position according to the node displacement vector, aligning the main direction of the material at the node position with the main stress direction, and optimizing the direction and the density of the material according to a full stress criterion.
In this step, the optimization problem can be stated as
find t bj ,n bj
min V
s.t.|σ bj |≤σ p
Where V is the structural volume, σ bj Is the principal stress, σ, of the node position p Is the material allowable stress.
And optimizing the density of the material at the position of the joint j according to a full stress criterion method:
Figure GDA0003866574780000072
in the formula the superscript i denotes the iteration index,
Figure GDA0003866574780000073
is the principal stress, σ p Is the material allowable stress.
In order to avoid singular structural rigidity matrix, the material density is not limited to be too low
Figure GDA0003866574780000074
Figure GDA0003866574780000075
The alignment of the principal direction of the material at the location of the junction with the principal stress direction is shown below
Figure GDA0003866574780000076
In the formula, m bj Indicating the principal stress direction.
5) And (3) judging whether the maximum values of the relative variation and the direction absolute variation of the material density of each node position are small enough in the two continuous iteration processes, returning to the step 2) if not, carrying out structural analysis and optimization again, and entering the step 6) if yes.
Specifically, whether the maximum value of the relative variation and the maximum value of the direction absolute variation of the material density at each node position in the two successive iterations is small enough is checked, and the maximum value is represented as follows:
Figure GDA0003866574780000081
wherein δ is a preset value, and | ∞ is an ∞ -norm.
Density t of material at arbitrary position (x, y, z) b And interpolating by using a shape function according to the density of the material at the node position to obtain:
Figure GDA0003866574780000082
6) And calculating the thickness of the material and the position of the middle surface of the curved shell structure along each vertical ridge line to form an optimized middle surface of the spatial curved shell structure.
The material thickness is calculated from the following formula,
Figure GDA0003866574780000083
1 st of the formula
Figure GDA0003866574780000084
All units in the vertical direction representing the same horizontal position are accumulated; t is t b Is the density of the material at any position (x, y, z),
Figure GDA0003866574780000085
position z of the face in the curved shell structure c (x j ,y j ) Calculated from the centroid formula:
Figure GDA0003866574780000086
thereby forming an optimized curved shell structure midplane.
Examples of applications are:
the structure within the rectangular planar domain is subjected to evenly distributed vertical direction loads. And the topological optimization design of the curved shell structure with the most saved materials under the stress constraint is met. As shown in FIG. 2, only 1/4 of the structure was actually analyzed due to symmetry.
The optimization steps are as follows:
1) The design domain upper space is divided into 12 × 12=1728 eight-node cubic space units. All nodes below the four edges of the boundary impose displacement constraints.
2) And distributing the vertical load according to the distribution of the material density along the vertical direction.
3) And forming a structural rigidity equation and solving to obtain a node displacement column vector.
4) And calculating the main stress direction and magnitude of each node position. The density of the material at the location of the nodes is varied according to the full stress criterion, aligning the material principal direction with the stress principal direction. See fig. 3.
5) And verifying the convergence condition. And if not, returns to 2.
6) And calculating the material volume of the structure, and calculating the centroid position of the material along the vertical direction. Connecting these centroids constitutes an optimized surface, see fig. 4, 5.
7) And optimizing the visual output of the result.
The above description is only an embodiment of the present invention, but the design concept of the present invention is not limited thereto, and any insubstantial modifications made by using the design concept should fall within the scope of infringing the present invention.

Claims (8)

1. A method for optimizing the shape of a spatial curved shell structure is characterized by comprising the following steps:
1) Setting a design domain, dividing the design domain by using a limited unit, and initializing the direction and density of a material at a node;
2) Distributing the load to each node in the design domain according to the material density distribution;
3) Establishing a structural rigidity matrix of a finite unit, assembling into a rigidity equation, introducing displacement constraint, solving the rigidity equation to obtain a node displacement vector, wherein the structural rigidity matrix is as follows:
Figure FDA0003882716110000011
in the formula, S e Representing a set of nodes belonging to a finite element e, t bj Which represents the density of the material and is,
Figure FDA0003882716110000012
e is the modulus of elasticity, V e Representing the range of each finite element, B is the set matrix, T represents the transpose, A r Is a constant matrix independent of node location, r =1,2.. 6,g r (n bj ) About the direction of the material
Figure FDA0003882716110000013
As a function of (a) or (b),
Figure FDA0003882716110000014
and
Figure FDA0003882716110000015
representing the components of the material direction vector on each coordinate axis; when a regular finite element is used, the matrix is a constant matrix;
the stiffness equation is as follows:
KU＝F
wherein, F is a load column vector, and U is a node displacement column vector;
principal stress is determined by solving the eigenvector m of the equation bj And the eigenvalue σ bj Obtaining:
Figure FDA0003882716110000016
the element of the 1 st matrix in the formula is a stress component, and is calculated by the following formula
[σ x σ y σ z τ yz τ zy τ xy ] T ＝DBU e
D is an elastic matrix, B is a set matrix, U e Is a node displacement column vector of finite elements;
solving the stiffness equation to obtain a node displacement column vector U,
the resulting structure volume was as follows:
Figure FDA0003882716110000017
in the formula N j Is a shape function of a finite element;
4) Calculating the main stress of the node position according to the node displacement vector, aligning the main direction of the material at the node position with the main stress direction, and optimizing the direction and the density of the material according to the full stress criterion;
5) Judging whether the maximum values of the relative variation and the direction absolute variation of the material density of each node position in the two continuous iteration processes are smaller than or equal to a preset value, if not, returning to the step 2), carrying out structural analysis and optimization again, and if so, entering the step 6);
6) And calculating the thickness of the material and the position of the middle surface of the curved shell structure to form an optimized middle surface of the spatial curved shell structure.
2. The method for optimizing the shape of the spatial curved shell structure according to claim 1, wherein in the step 1), the design domain is subdivided by eight-node rectangular units, and all edges of the eight-node rectangular units are kept in a vertical direction to form a plurality of vertical lines.
3. The method for optimizing the shape of the spatial curved shell structure according to claim 1, wherein the direction and density of the initialization material at the nodes are as follows: three main direction initial values of the three orthogonal materials at any node j position of the material are set as coordinate axis directions:
Figure FDA0003882716110000021
the initial density of the material in the three material main axis directions is t bj ＝0.2,b＝1,2,3。
4. The method for optimizing the shape of the spatial curved shell structure according to claim 1, wherein the vertical load distributed in the vertical direction in the step 2) is distributed to the edges of the finite elements according to a virtual work principle, and the expression is as follows:
Figure FDA0003882716110000022
integral field V in formula e Represent each oneRange of finite elements, N p (x, y, z) is a shape function, q (x, y) is a distribution function of vertical load in the horizontal direction, and x, y, z represents a spatial coordinate position.
5. The method for optimizing the shape of the spatial curved shell structure according to claim 1, wherein in the step 4), the density of the material at the position of the joint j is optimized according to a full stress criterion method
Figure FDA0003882716110000023
In the formula the superscript i denotes the iteration index,
Figure FDA0003882716110000024
is the principal stress, σ p Is the material allowable stress; in order to avoid singular of the structural rigidity matrix, the material density is limited
Figure FDA0003882716110000025
Figure FDA0003882716110000026
6. The method for optimizing the shape of a space curved shell structure according to claim 1, wherein in the step 4), the main direction of the material at the node position is aligned with the main stress direction as follows
Figure FDA0003882716110000027
In the formula, m bj Indicating the principal stress direction.
7. The method for optimizing the shape of a spatial curved shell structure according to claim 1, wherein in the step 5), it is checked whether the maximum value of the relative variation and the direction absolute variation of the material density at each node position is sufficiently small in the two successive iterations, which is specifically expressed as follows:
Figure FDA0003882716110000031
wherein δ is a preset value, | | |. | non-calculation ∞ Is an infinite-norm.
8. The method for optimizing the shape of a space curved shell structure according to claim 1, wherein the thickness of the material is calculated by the following formula,
Figure FDA0003882716110000032
1 st of the formula
Figure FDA0003882716110000033
All units in the vertical direction representing the same horizontal position are accumulated; t is t b Is the density of the material at any position (x, y, z),
Figure FDA0003882716110000034
the position z of the surface in the curved shell structure c (x j ,y j ) Calculated from the centroid formula:
Figure FDA0003882716110000035
thereby forming an optimized curved shell structure midplane.
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