CN110245410B - Multi-parametric variable-based topological optimization design method for thermoelastic structure of multiphase material - Google Patents
Multi-parametric variable-based topological optimization design method for thermoelastic structure of multiphase material Download PDFInfo
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Abstract
The invention provides a multi-parameterization variable-based topological optimization method for a multiphase material thermoelastic structure, which is characterized in that a plurality of parameterized B-spline spaces are used for modeling and describing a design domain, control points of each B-spline space are used as design variables of an optimization problem, each B-spline space corresponds to a phase of material, and control point grids, control point grids and structural background grids in which the design variables are located are independent of one another, so that fewer design variables can be used for corresponding to a finer background grid. The design variable adopted by the method does not depend on a background grid, and the method has high-order continuity, and the high-order continuity of the B spline space enables the method to have better global convergence, and numerical problems such as intermediate materials, checkerboard phenomena and the like can be automatically avoided without other means.
Description
Technical Field
The invention relates to a topological optimization design method for a multiphase material thermoelastic structure, in particular to a topological optimization method for a multiphase material thermoelastic structure based on B-spline form multi-parametric variables.
Background
Multiphase material structures are structures that are composed of a variety of materials of different properties and functions. In engineering design, multiphase material structures composed of various solid materials and cavities are common. The multiphase material topology optimization method is a method for designing the dosage, shape and hollow space occupancy of each phase material to form a reasonable layout by aiming at seeking the optimal performance (such as the maximum integral rigidity) of a structure under the given boundary condition and the structural mass or volume constraint. The thermoelastic structure is a structure that expands or contracts due to the properties of a material itself under a thermal load, thereby generating a mechanical stress therein. The topological optimization of the thermoelastic structure of the multiphase material has higher flexibility and complexity compared with the topological optimization of a single material, and the performance of the structure is improved from more dimensions. The thermal coupling effect generated by the thermoelastic structure when subjected to mechanical and thermal loads also puts higher demands and challenges on topology optimization.
The document "A mass constraint formation for structural coordination with multiple phase materials. Tong Gao, Weihong Zhang. int.J.Numer. meth. Engng 2011; 774-. The literature interpolates properties of multiphase materials, such as young's modulus, volume, density, etc., using a plurality of cell-based pseudo-density variables, and optimizes the shape and material properties of the structure via a gradient optimization algorithm. The method analyzes two typical material interpolation models: a recursive material interpolation model (RMMI) and a peer-to-peer material interpolation model (UMMI) and comparing the calculation accuracy thereof under the mass-constrained and volume-constrained conditions, respectively. By comparison, a peer-to-peer material interpolation model (UMMI) can obtain a better structure, and the advantages of the UMMI are more remarkable under the quality constraint. However, the methods described in the literature use discrete cell-based design variables, have poor continuity between cells, are numerous, and are prone to inter-material and checkerboard phenomena, which must be improved by additional numerical processing.
Disclosure of Invention
The invention provides a multi-parametric variable-based topological optimization method for a multiphase material thermoelastic structure, which aims to solve the problems of poor variable continuity and easiness in occurrence of intermediate materials and checkerboard phenomena in the traditional multiphase material topological optimization method and is applied to the thermoelastic structure.
The method models and describes a design domain using a plurality of parameterized B-spline spaces. Unlike conventional pseudo-density methods, the present method replaces discrete cell-based design variables with higher-order continuous variables. The control points of each B-spline space are used as design variables of the optimization problem, and each B-spline space corresponds to one phase of material. The control point grids where the design variables are located, and the control point grids and the structural background grids are independent of each other. Fewer design variables may be used for finer background grids. For the multi-phase material optimization problem, the method combines a B-spline space with a peer-to-peer material interpolation model (UMMI) and optimizes under corresponding volume constraint and mass constraint. In order to accelerate convergence, the method adds numerical penalties of the traditional SIMP form and the RAMP form. Compared with the design method in the background technology, the design variables adopted by the method do not depend on the background grid, the method has high-order continuity, the high-order continuity of the B spline space enables the method to have better global convergence, and numerical problems such as intermediate materials, checkerboard phenomena and the like can be automatically avoided without other means.
Based on the principle, the technical scheme of the invention is as follows:
the multi-parametric variable based multiphase material thermoelastic structure topology optimization design method is characterized by comprising the following steps of: the method comprises the following steps:
step 1: determining a dimension l of a design Domain x ×l y The number m of multiphase materials, and determining the degree p of m B-spline spaces and the respective control point grid size nx k ×ny k K is 1.., m; establishing a parameter area corresponding to the design domain, setting the parameter area (xi, eta), and setting the design domain (x, y) to be in the scope of 0, l x ]×[0,l y ]Then there is
Step 2: establishing m B-spline spaces, and pseudo-density values rho of any point in the spaces (k) (xi, eta) is represented by the formula
Is obtained wherein P is ij (k) Representing the pseudo-density value, P, at the k-th grid control point min Is a set minimum, N i,p (ξ)，N j,p (η) is a P-th order B-spline shaped function;
and step 3: embedding a design domain into a regular rectangular region omega, dividing a regular quadrilateral grid, and establishing a finite element model; given a level set function phi of a design domain, calculating a stiffness matrix of each cell:
wherein omega e Representing the area of the unit, wherein H (phi) is a Heaviside function, B is a strain-displacement matrix, and D (rho) is a unit elastic matrix; calculated from the peer interpolation model:
in the formula D i An elastic matrix representing the ith material, and χ (ρ) representing a pseudo-density penalty function;
for thermoelastic structures, it is also necessary to calculate the thermal load F Th ：
F Th ＝ΔT∫ Ω B T ·β·H(φ)dΩ
Wherein Δ T represents the temperature change of the structure, β is the thermal stress coefficient vector of the structure, calculated by the peer-to-peer material interpolation model:
wherein beta is i ＝α i ·D i ·Φ，α i The coefficient of thermal expansion for the ith material, Φ is a constant vector: two-dimensional where Φ is [ 110 ]] T (ii) a Three-dimensional medium phi ═ 111000] T ；
And 4, step 4: assembling the rigidity matrix of each unit into a structural integral rigidity matrixK, applying boundary conditions and loads on the finite element model, wherein the loads comprise set mechanical loads F Me With the thermal load F obtained in step 3 Th Establishing a mechanical model K (rho) of the macrostructure (k) )·U＝F Me +F Th And solving a node displacement vector U;
and 5: selecting control point data of m B spline spaces as a design variable, selecting structural flexibility C as an optimization target, setting an initial value and a variation range of the design variable as a constraint function, and establishing an optimization model of a multi-phase material topology optimization problem:
in the formula g Vk Represents a volume fraction of
Representing a given upper limit of the volume fraction, there is
Step 6: and 5, carrying out optimization solution on the optimization model established in the step 5 to obtain an optimization result.
Further, in a preferred embodiment, the method for topologically optimally designing the thermoelastic structure of the multi-phase material based on the multi-parametric variables is characterized in that: the B spline functions in the step 2 all adopt the following forms, wherein N i,p (xi) is
In the formula, xi i ∈{ξ 1 ,ξ 2 ,...,ξ nxk+p+1 Is nx uniformly distributed over (0,1) k -p-1 nodes and p +1 repeating nodes at ξ ═ 0 and ξ ═ 1, respectively; and for N j,p (η), i in the above form is replaced by j, ξ is replaced by η, nx k Is replaced by ny k 。
Further, in a preferred embodiment, the method for topologically optimally designing the thermoelastic structure of the multi-phase material based on the multi-parametric variables is characterized in that: in the step 3, the pseudo-density penalty function takes two forms of SIMP and RAMP according to the working condition:
where pn is a penalty factor.
Advantageous effects
Compared with the prior art, the invention has the beneficial effects that: the method adopts a plurality of independent B-spline spaces as basic design elements for topology optimization, so that design variables do not depend on background grids any more, and the fine grids are controlled by less variables. And the continuous field of the B-spline space can ensure that the structure has a continuous profile and material distribution. In addition, the invention can automatically avoid the numerical defects such as gray level units, intermediate materials, checkerboard phenomena and the like caused by discrete variables due to the adoption of continuous variables, has better convergence, and can obtain better results.
Additional aspects and advantages of the invention will be set forth in part in the description which follows and, in part, will be obvious from the description, or may be learned by practice of the invention.
Drawings
The above and/or additional aspects and advantages of the present invention will become apparent and readily appreciated from the following description of the embodiments, taken in conjunction with the accompanying drawings of which:
FIG. 1 is a schematic diagram of the B-spline space multi-parameterization of the method of the present invention.
FIG. 2 is a schematic drawing of the control point grid (left) and B-spline space (right) for the method of the present invention.
FIG. 3 is a schematic diagram of the design area and operating conditions of the method of the present invention. An MBB beam structure is shown.
FIG. 4 is a graph of the optimization results of the B-spline multi-parameterization of the method of the present invention, showing three different materials together, with a clear interface between the materials.
FIG. 5 shows three B-spline surfaces in the optimization result of the method of the present invention. The three curved surfaces respectively represent the distribution range of the three materials, the white part of the curved surface represents the materials, and the black part represents the holes.
Detailed Description
The following detailed description of embodiments of the invention is intended to be illustrative, and not to be construed as limiting the invention.
Refer to fig. 3 to 5. In this embodiment, a multiphase material structure topology optimization design is performed for an MBB beam structure, the size of the design area is 60mm × 20mm, the left side of the area is constrained in the horizontal direction, and the lower right corner is constrained in the vertical direction. The upper left corner of the zone is subjected to a vertically downward concentrated force F of 100N, free of thermal loading. Three different materials are adopted for design, and the Young modulus and the Poisson ratio of the three candidate materials are respectively E 1 ＝70Pa,E 2 ＝120Pa,E 3 ＝210Pa,ν 1 ＝ν 2 ＝0.34,ν 3 ＝0.3。
Step 1: determining that the number of required B sample band spaces is m-3, selecting the number of times of B sample bands as p-5, and controlling the grid size of each B sample band control point to be 60 multiplied by 20; referring to FIG. 1, a parameter area corresponding to a design field is established, the parameter area (xi, eta) belongs to [0,1] × [0,1], the design field (x, y) belongs to [0,60] × [0,20], and then
Step 2: establishing 3B spline spaces, and setting P min ＝10 -5 The value rho of pseudo density at any point in space is the lower bound of the control point, aiming at avoiding the numerical problem caused by the fact that the pseudo density is equal to zero (k) (xi, eta) is represented by the formula
Is obtained, wherein P ij (k) Representing the pseudo density value, N, at the kth grid control point i,p (ξ)，N j,p (η) is a p-th order B-spline function, in the form of where N i,p (xi) is
In the formula, xi i ∈{ξ 1 ,ξ 2 ,...,ξ nxk+p+1 Is nx uniformly distributed over (0,1) k -p-1 nodes and p +1 repeating nodes at ξ ═ 0 and ξ ═ 1, respectively; and for N j,p (η), i in the above form is replaced by j, ξ is replaced by η, nx k Is replaced by ny k 。
And step 3: embedding a design domain into a regular rectangular region omega, dividing a regular quadrilateral grid, and establishing a finite element model; the level set function phi is given for the design domain. In this embodiment, since the design domain itself is quite regular, it is not necessary to calculate the level set function Φ.
Calculating a stiffness matrix for each cell:
wherein omega e Representing the area of the unit, wherein H (phi) is a Heaviside function, B is a strain-displacement matrix, and D (rho) is a unit elastic matrix; calculated from the peer interpolation model:
in the formula D i An elastic matrix representing the ith material, and χ (ρ) representing a pseudo-density penalty function; in the embodiment, the method comprises the following steps:
D(ρ)＝χ(ρ (1) )(1-χ(ρ (2) ))(1-χ(ρ (3) ))+(1-χ(ρ (1) ))χ(ρ (2) )(1-χ(ρ (3) ))+(1-χ(ρ (1) ))(1-χ(ρ (2) ))χ(ρ (3) )
the pseudo-density penalty function takes two forms of SIMP and RAMP according to working conditions:
where pn is a penalty factor. Considering that the working condition in this embodiment is a static working condition, an SIMP form χ (ρ) ρ is selected with a penalty coefficient pn of 3 3 . And the variable ρ (1) -ρ (3) This can be obtained from step 2.
And 4, step 4: and assembling the rigidity matrix of each unit into a structural integral rigidity matrix K, applying boundary conditions and loads F on a finite element model, establishing a mechanical model K (rho). U.F of a macroscopic structure, and solving a node displacement vector U.
And 5: selecting control point data of m B spline spaces as design variables, selecting structure flexibility C as an optimization target, using volume fraction performance indexes of the structure as constraint functions, and setting initial values of the design variables to be rho (1) ＝ρ (2) ＝ρ (3) 0.16, the upper and lower limits of the variable are respectively 10 -5 And 1, establishing an optimization model of the multiphase material topology optimization problem:
in the formula g V1 ,g V2 And g V3 Representing the volume fractions of material 1, material 2 and material 3, respectively. The volume fractions of the three materials are not limited to be more than 16.67 percent, then
Step 6: BOSS-Quattro on optimized design platform TM And (5) carrying out optimization solution on the optimization model established in the step (5) by using a GCMMA optimization algorithm to obtain an optimization result.
The material distribution and convergence curve of the optimized structure are shown in fig. 4. The top view of the B-spline surface for the three materials is shown in fig. 5. The multiphase material topology optimization design method based on multi-parametric variables can obtain clear and smooth material boundaries and structural configurations, the B-spline surface corresponding to each material has high-order continuity, the convergence process is stable and rapid, and the numerical problems of gray level units and the like do not occur in the whole optimization process. The result shows that the multiphase material thermoelastic topological optimization design method based on multi-parametric variables can obtain clear and smooth material boundaries and structure configurations while improving the structural rigidity, and solves the problems that the design method in the background art is poor in variable continuity and easy to generate intermediate materials and checkerboard phenomena.
Although embodiments of the present invention have been shown and described above, it is understood that the above embodiments are exemplary and should not be construed as limiting the present invention, and that variations, modifications, substitutions and alterations can be made in the above embodiments by those of ordinary skill in the art without departing from the principle and spirit of the present invention.
Claims (3)
1. A multi-parametric variable based topological optimization design method for a multi-phase material thermoelastic structure is characterized by comprising the following steps of: the method comprises the following steps:
step 1:determining a dimension l of a design Domain x ×l y The number m of multiphase materials, and determining the degree p of m B-spline spaces and the respective control point grid size nx k ×ny k K is 1.., m; establishing a parameter area corresponding to the design domain, setting the parameter area (xi, eta), and setting the design domain (x, y) to be in the scope of 0, l x ]×[0,l y ]Then there is
Step 2: establishing m B-spline spaces, and pseudo-density values rho of any point in the spaces (k) (xi, eta) is represented by the formula
Is obtained, wherein P ij (k) Representing the pseudo-density value, P, at the k-th grid control point min Is a set minimum, N i,p (ξ)，N j,p (η) is a P-th order B-spline shaped function;
and step 3: embedding a design domain into a regular rectangular region omega, dividing a regular quadrilateral grid, and establishing a finite element model; given a level set function phi of a design domain, calculating a stiffness matrix of each cell:
wherein Ω is e Representing the area of the unit, wherein H (phi) is a Heaviside function, B is a strain-displacement matrix, and D (rho) is a unit elastic matrix; calculated from the peer interpolation model:
in the formula D i An elastic matrix representing the i-th material,χ (ρ) represents a pseudo density penalty function;
for thermoelastic structures, it is also necessary to calculate the thermal load F Th ：
Wherein Δ T represents the temperature change of the structure, β is the thermal stress coefficient vector of the structure, calculated by the peer-to-peer material interpolation model:
wherein beta is i ＝α i ·D i ·Φ，α i The coefficient of thermal expansion for the ith material, Φ is a constant vector: two-dimensional where phi is [ 110 ═] T (ii) a Three-dimensional medium phi ═ 111000] T ；
And 4, step 4: assembling the rigidity matrix of each unit into a structural integral rigidity matrix K, and applying boundary conditions and loads on a finite element model, wherein the loads comprise set mechanical loads F Me With the thermal load F obtained in step 3 Th Establishing a mechanical model K (rho) of the macrostructure (k) )·U＝F Me +F Th And solving a node displacement vector U;
and 5: selecting control point data of m B spline spaces as a design variable, selecting structural flexibility C as an optimization target, setting an initial value and a variation range of the design variable as a constraint function, and establishing an optimization model of a multi-phase material topology optimization problem:
in the formula g Vk Represents a volume fraction of
Representing a given upper limit of the volume fraction, there is
Step 6: and 5, carrying out optimization solution on the optimization model established in the step 5 to obtain an optimization result.
2. The topological optimization design method for the multiphase material thermoelastic structure based on the multi-parametric variables according to claim 1, characterized in that: the B-spline functions in step 2 all adopt the following forms, wherein N i,p (xi) is
In the formula, xi i ∈{ξ 1 ,ξ 2 ,...,ξ nxk+p+1 Is nx uniformly distributed over (0,1) k -p-1 nodes and p +1 repeating nodes at ξ ═ 0 and ξ ═ 1, respectively; and for N j,p (η), i in the above form is replaced by j, ξ is replaced by η, nx k Is replaced by ny k 。
3. The topological optimization design method for the multiphase material thermoelastic structure based on the multi-parametric variables according to claim 1, characterized in that: in the step 3, the pseudo-density penalty function takes two forms of SIMP and RAMP according to the working condition:
where pn is a penalty factor.
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