Topological optimization method of non-orthogonal truss continuum structure
By introducing low-density orthogonal rods as base materials into the truss-like structure, the non-orthogonal truss material model is solved, and a more flexible material distribution description and minimum flexibility optimization under multiple operating conditions is achieved.
Patent Information
- Application Number
- CN202411800605.9
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- Filing Date
- 2024-12-09
- Publication Date
- 2025-03-18
AI Technical Summary
Traditional orthogonal truss structural models tend to cause singular stiffness problems when dealing with non-orthogonal rods, and cannot effectively describe the distribution field of general truss materials.
The topological optimization method based on general truss material model and mathematical planning solution method is adopted. By constructing a non-orthogonal truss material model, low-density orthogonal rods are introduced as the basis material, and the singular stiffness problem caused by the overlap of non-orthogonal rods is solved.
Effectively lift the limitations on the orthogonality of the rod, enhance the description ability of the material model, avoid the problem of singular stiffness, and achieve the optimization of minimum flexibility under multiple operating conditions.
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Figure CN119647018A_ABST
Abstract
Description
Technical Field
[0001] The invention relates to the technical field of structural optimization, and in particular to a topology optimization method for a non-orthogonal truss-like continuum structure. Background Art
[0002] When establishing a general truss-like material model for a truss-like continuum, the traditional method is to establish an orthogonal truss-like material model. Figure 1 As shown in the figure, the so-called orthogonal framework material model is: under plane stress state, it is understood as a structure constructed by 2D materials with special properties in the sense of continuous medium, specifically: (1) a structure made of materials with only two orthogonal directions (corresponding to the principal strain directions); (2) the directional density is proportional to the absolute value of the corresponding principal stress; 3) the total density is equal to the sum of the directional densities. As can be seen from the above, it is not applicable to general material separation fields. Summary of the invention
[0003] 1. Technical problems to be solved:
[0004] In view of the above technical problems, the present invention provides a topology optimization method for a non-orthogonal truss-like continuum structure. On the basis of the traditional topology optimization method for a truss-like structure, a topology optimization method based on a general truss-like material model and a mathematical programming solution method is constructed, while solving the problem of stiffness singularity that may be caused by the overlap of non-orthogonal rods.
[0005] 2. Technical solution:
[0006] A topology optimization method for a non-orthogonal truss-like continuum structure, characterized by comprising:
[0007] Step 1: Taking the density and direction of the rods at the nodes as design variables, two sets of non-orthogonal truss-like material models are established based on the orthogonal truss-like material model;
[0008] Step 2: Based on the two sets of non-orthogonal truss-like material models constructed, the initial design domain is divided into finite element meshes to establish a finite element calculation model; initial values are assigned to the design variables and the initial number of iterations is set; according to the finite element calculation model, the structure is subjected to finite element analysis to obtain the node displacement, strain and stress of the truss-like continuum;
[0009] Step 3: Use the power function to dynamically update the finite element calculation model to obtain optimized model parameters;
[0010] Step 4: Based on the optimized model, the moving asymptote mathematical optimization algorithm is used to effectively solve the classic optimization problem of minimum flexibility under multiple working conditions.
[0011] Furthermore, step one specifically includes: S11: the general truss material model is regarded as the superposition of two groups of orthogonal members, the densities of the two groups of members are t1 and t2 respectively; the directions of the members are a1 and a2 respectively; the density of one member of each group of orthogonal members is 0; then the elastic matrix of the two member directions is:
[0012]
[0013] In the above formula, E is the Young's modulus of the material; RG is the parameter that controls the shear stiffness of the material in the traditional structural relationship, and the value of different material models is usually different;
[0014] S12: The elastic matrix of the two-way non-orthogonal truss material model is obtained by combining the elastic matrix of the orthogonal truss material model; wherein the elastic matrix of the orthogonal truss material model along the coordinate axis is:
[0015]
[0016] In the above formula, a is the angle between the material principal axis and the rectangular coordinate system in the orthogonal truss material model; T is the coordinate transformation matrix; where:
[0017]
[0018] According to equations (1) and (2), the elastic matrix based on the two-dimensional non-orthogonal truss material model is obtained as follows:
[0019]
[0020] After sorting:
[0021]
[0022] In the above formula, gr is the element in the matrix g:
[0023] g(α)=[R G cos4α R G sin4α cos2α sin2α R G 1];
[0024] The constant matrix A is:
[0025]
[0026] S13: Add two low-density orthogonal bars as base materials in the two-dimensional non-orthogonal material model; the elastic matrix of the base material is recorded as:
[0027] D0(t0)=Et0diag[1 1 0.5] (5);
[0028] In the above formula, t0 is the density of the two foundation materials; the two foundation materials are isotropic materials, and the stiffness distribution along all directions is the same;
[0029] S14: Ignore the distribution angle of the base material bar and directly superimpose it on the elastic matrix of the non-orthogonal material model; the elastic matrix of the improved non-orthogonal material model is:
[0030]
[0031] Furthermore, step 2 specifically includes:
[0032] S21: For the regular design domain, bilinear rectangular isoparametric elements are used for discretization, and the structural mechanical response is solved by finite element analysis; the design domain is divided into regular rectangular finite elements; two groups of non-orthogonal truss-like members are distributed on each unit node; the dense curve in the unit represents the distribution direction of the truss-like members;
[0033] S22: The elasticity matrix of any node in the unit is as follows:
[0034]
[0035] In the above formula, e is the unit number; ξ, η are local parameter coordinates, and Se is the set of node numbers around the unit; represents the material elastic matrix at the th node;
[0036] Substitute the following unit stiffness matrix in the finite element theory into equation (7):
[0037]
[0038] In the above formula, Ωe represents the integration domain of the unit;
[0039] The following formula (8) is obtained:
[0040]
[0041] in:
[0042]
[0043] S23: Elastic plane stress is as follows:
[0044]
[0045] In the above formula, i1 and i2 are two coordinate rotation indices, and h is the plane thickness;
[0046]
[0047] in:
[0048]
[0049] S24: Assemble to obtain the overall stiffness matrix, as follows:
[0050]
[0051] Furthermore, in step 3, the power function form used is as follows:
[0052] R G =(1-l5) q ;
[0053] In the above formula, the independent variable l5 is the average rate of change of the objective function for the last five times, l min =10 -n is the lower limit of the average rate of change; q is the penalty factor; definition:
[0054] l5=max{min{l5,1},l min}.
[0055] 3. Beneficial effects:
[0056] Based on the traditional truss-like structure topology optimization method, this method constructs a set of orthogonal bars with basic material functions to solve the stiffness singularity problem that may be caused by the overlap of non-orthogonal bars, removes the restriction on the orthogonality of bars, and enhances the description ability of the material model. On this basis, a set of low-density orthogonal bars are introduced as the base material to solve the stiffness singularity problem caused by the overlap of bars in the material model. BRIEF DESCRIPTION OF THE DRAWINGS
[0057] Figure 1 For the existing orthogonal truss material model;
[0058] Figure 2 A generalized non-orthogonal bar material model constructed for this method;
[0059] Figure 3 This is the regular design domain designed in this method, where the left figure is the finite element mesh division; the right figure is the distribution field of the rod in the finite element;
[0060] Figure 4 Flowchart of this method. DETAILED DESCRIPTION
[0061] The present invention will be described in detail below with reference to the accompanying drawings.
[0062] like Figure 4 As shown in FIG. 1 , it is the overall flow chart of this method. Figure 2As shown in the figure, the schematic diagram of the generalized non-orthogonal member material model constructed by this method is shown; in order to describe a more general material distribution field, a general truss-like material model with variable shear stiffness is designed. It is assumed that there are two groups of members, and the member densities are t1 and t2 respectively; the member direction angles are a1 and a2 respectively; the non-orthogonal member model can be regarded as the superposition of two groups of orthogonal members, and the density of one member in each group of orthogonal members is zero. Therefore, the elastic matrices along the two member directions can be expressed as formula (1) respectively:
[0063]
[0064] In the above formula, E is the Young's modulus of the material; RG is the parameter that controls the shear stiffness of the material in the traditional structural relationship, and the value of different material models is usually different.
[0065] The elastic matrix of the orthogonal truss material model is combined with the elastic matrix of the bidirectional non-orthogonal truss material model to obtain the elastic matrix of the bidirectional non-orthogonal truss material model; the elastic matrix of the orthogonal truss material model along the coordinate axis is:
[0066]
[0067] In the above formula, a is the angle between the material principal axis and the rectangular coordinate system in the orthogonal truss material model; T is the coordinate transformation matrix; where:
[0068]
[0069] According to equations (1) and (2), the elastic matrix based on the two-dimensional non-orthogonal truss material model is obtained as follows:
[0070]
[0071] After sorting:
[0072]
[0073] In the above formula, gr is the element in the matrix g:
[0074] g(α)=[R G cos4α R G sin4α cos2α sin2α R G 1];
[0075] The constant matrix A is:
[0076]
[0077] In the two-dimensional non-orthogonal material model, two low-density orthogonal bars are added as base materials; the elastic matrix of the base material is recorded as:
[0078] D0(t0)=Et0diag[1 1 0.5] (5);
[0079] In the above formula, t0 is the density of the two foundation materials; the two foundation materials are isotropic materials, and the stiffness distribution along all directions is the same;
[0080] Ignore the distribution angle of the base material bar and directly superimpose it on the elastic matrix of the non-orthogonal material model; the elastic matrix of the improved non-orthogonal material model is:
[0081]
[0082] After establishing the truss-like material model, the regular design domain is discretized using bilinear rectangular isoparametric elements, and the structural mechanical response is solved by finite element analysis. Figure 3 As shown in the left figure of , the design domain is divided into regular rectangular finite elements. There are two groups of non-orthogonal frame-like members distributed on each unit node, such as Figure 3 As shown in the right figure, t bj represents the rod density, a bj Indicates the direction of the member; b = 1, 2 indicates two groups of member indexes, j = 1, 2, ..., nd indicates the node number index, and nd, indicates the total number of nodes. The dense curve in the unit indicates the distribution direction of the truss-like members.
[0083] The elasticity matrix of any node in the unit is as follows:
[0084]
[0085] In the above formula, e is the unit number; ξ, η are local parameter coordinates, and Se is the set of node numbers around the unit; represents the material elastic matrix at the th node;
[0086] Substitute the following unit stiffness matrix in the finite element theory into equation (7):
[0087]
[0088] In the above formula, Ωe represents the integration domain of the unit;
[0089] The following formula (8) is obtained:
[0090]
[0091] in:
[0092] The elastic plane stress is given by:
[0093]
[0094] In the above formula, i1 and i2 are two coordinate rotation indices, and h is the plane thickness;
[0095]
[0096] in:
[0097]
[0098] The overall stiffness matrix is obtained by assembly as follows:
[0099]
[0100] The power function is used to dynamically update the finite element calculation model to obtain the optimized model parameters;
[0101] The power function used is as follows:
[0102] R G =(1-l5) q ;
[0103] In the above formula, the independent variable l5 is the average rate of change of the objective function for the last five times, l min =10 -n is the lower limit of the average rate of change; q is the penalty factor; definition:
[0104] l5=max{min{l5,1},l min}.
[0105] Based on the optimized model, the moving asymptote mathematical optimization algorithm is used to effectively solve the classical optimization problem of minimum flexibility under multiple working conditions.
[0106] Although the present invention has been disclosed as above in terms of preferred embodiments, they are not intended to limit the present invention. Anyone skilled in the art can make various changes or modifications without departing from the spirit and scope of the present invention. Therefore, the scope of protection of the present invention should be based on the scope of protection defined by the claims of this application.
Claims
1. A topology optimization method for a non-orthogonal truss-like continuum structure, characterized in that: include: Step 1: Taking the density and direction of the rods at the nodes as design variables, two sets of non-orthogonal truss-like material models are established based on the orthogonal truss-like material model; Step 2: Based on the two sets of non-orthogonal truss-like material models, the initial design domain is meshed to establish a finite element calculation model; the design variables are assigned initial values and the initial number of iterations is set; According to the finite element calculation model, a finite element analysis is performed on the structure to obtain the node displacement, strain and stress of the truss-like continuum; Step 3: Use the power function to dynamically update the finite element calculation model to obtain optimized model parameters; Step 4: Based on the optimized model, the moving asymptote mathematical optimization algorithm is used to effectively solve the classic optimization problem of minimum flexibility under multiple working conditions.
2. The topology optimization method of a non-orthogonal truss-like continuum structure according to claim 1, characterized in that: Step 1 specifically includes: S11: The general truss material model is regarded as the superposition of two groups of orthogonal members, the densities of the two groups of members are t1 and t2 respectively; the directions of the members are a1 and a2 respectively; the density of one member in each group of orthogonal members is 0; then the elastic matrix of the two member directions is: In the above formula, E is the Young's modulus of the material; RG is the parameter that controls the shear stiffness of the material in the traditional structural relationship, and the value of different material models is usually different; S12: Combining the elastic matrix of the orthogonal truss material model, an elastic matrix based on the two-way non-orthogonal truss material model is obtained; wherein the elastic matrix of the orthogonal truss material model along the coordinate axis direction is: In the above formula, a is the angle between the material principal axis and the rectangular coordinate system in the orthogonal truss material model; T is the coordinate transformation matrix; where: According to equations (1) and (2), the elastic matrix based on the two-dimensional non-orthogonal truss material model is obtained as follows: After sorting: In the above formula, gr is the element in the matrix g: g(α)=[R G cos4α R G sin 4a cos2a sin2a R G 1]; The constant matrix A is: S13: Add two low-density orthogonal bars as base materials in the two-dimensional non-orthogonal material model; the elastic matrix of the base material is recorded as: D0(t0)=Et0diag[1 1 0.5] (5); In the above formula, t0 is the density of the two foundation materials; the two foundation materials are isotropic materials, and the stiffness distribution along all directions is the same; S14: Ignore the distribution angle of the base material bar and directly superimpose it on the elastic matrix of the non-orthogonal material model; the elastic matrix of the improved non-orthogonal material model is:
3. The topology optimization method of a non-orthogonal truss-like continuum structure according to claim 2, characterized in that: Step 2 specifically includes: S21: For the regular design domain, bilinear rectangular isoparametric elements are used for discretization, and the structural mechanical response is solved by finite element analysis; the design domain is divided into regular rectangular finite elements; two groups of non-orthogonal truss-like members are distributed on each unit node; the dense curve in the unit represents the distribution direction of the truss-like members; S22: The elasticity matrix of any node in the unit is as follows: In the above formula, e is the unit number; ξ, η are local parameter coordinates, and Se is the set of node numbers around the unit; represents the material elastic matrix at the th node; Substitute the following unit stiffness matrix in the finite element theory into equation (7): In the above formula, Ωe represents the integration domain of the unit; The following formula (8) is obtained: in: S23: Elastic plane stress is as follows: In the above formula, i1 and i2 are two coordinate rotation indices, and h is the plane thickness; in: S24: Assemble to obtain the overall stiffness matrix, as follows:
4. The topology optimization method of a non-orthogonal truss-like continuum structure according to claim 3, characterized in that: In step 3, the power function used is as follows: R G ＝(1-l5) q ； In the above formula, the independent variable l5 is the average rate of change of the objective function in the last five times, l min =10 -n is the lower limit of the average rate of change; q is the penalty factor; definition: l5=max{min{l5,1},l min }。