CN103488813A - Optimized design method for layout of multi-component structure system based on P norm - Google Patents

Abstract

The invention discloses an optimized design method for layout of multi-component structure system based on the P norm, which is used for solving the technical problems that the constraints in the existing method are excessive to cause difficult conduction or convergence of the optimization problem. The invention adopts the technical scheme that the optimized design method comprises the following steps that all component interference constraint equations are condensed into one or more constraint equations by using the P norm, all the component interference constraint equations are condensed into one or several constraint equations by the P norm, the obtained value of the P norm is close to a maximum value of all the constraint functions, and as a conservative estimation of the feasible region of the component interference problem, local or globally optimal solution can be found in the range of the original feasible region by optimization. The optimized design method has the advantages that the constraint condensation method is combined with the optimized design method with component synergy and topology on the basis of background grids and the density point technology, and the technical problems that the number of interference constraints in the traditional optimized design method with component synergy and topology in the background technology is excessive to cause difficult conduction or convergence of the optimization are solved, so that the optimization problem can be smoothly conducted and converged.

Description

Method for layout optimal design of multi-assembly structure system based on the P norm

Technical field

The present invention relates to a kind of method for layout optimal design of multi-assembly structure system.Be particularly related to a kind of method for layout optimal design of multi-assembly structure system based on the P norm.

Background technology

With reference to Fig. 1.This Design Mode of multicomponent structures system has been contained most industrial products, as aerospace flight vehicle, boats and ships, automobile and machinery etc.Due to its complicated duty status and harsh performance requirement, the mechanical property design problem of multicomponent structures system is particularly outstanding in aerospace flight vehicle structural design field.Because assembly and structure all have himself quality and rigidity, thus assembly put the comprehensive mechanical characteristic that has fundamentally determined structural system with this two aspects factor of location of configuration of structure.For the balance and stability that guarantees aircraft and avoid equipment or the damage of structure, need to carry out rational optimal design to these two kinds of location problems, this work is called the monolithic construction system by the collaborative topological design of multicomponent structures system.

With reference to Fig. 2.Document " Zhu J.H.; Beckers P.Zhang W.H.; On the multi-component layout design with inertial force.Journal of Computational and Applied Mathematics.2010; 234 (7): 2222-2230 " discloses a kind of method for layout optimal design of multi-assembly structure system, this method combines structural Topology Optimization technology and filling layout optimization technique, has realized that the location layout of the regional inner assembly of certain filling and support and connection version design simultaneously.Document discloses and has used limited envelope circule method to solve the assembly interference problem.Adopt the profile of limited envelope circle (three-dimensional micromodule is the envelope ball) this assembly of approximate description for the assembly of random appearance profile, if any two envelopes circle between two assemblies is not interfered, these two assemblies are not interfered.N assembly even arranged, i assembly (i=1,2 .n.. use) N _iindividual envelope circle is approximate, retrains this n assembly and does not interfere and need to have $\underset{i=1}{\overset{n-1}{\mathrm{\Σ}}}\underset{j=i+1}{\overset{n}{\mathrm{\Σ}}}{N}_i{N}_j$ Individual equation of constraint.

With reference to Fig. 3.The disclosed method for layout optimal design of multi-assembly structure system of document adopts background grid in conjunction with the density points technology, design domain is divided into to the background grid of unalterable rules, define density points in background grid, select assembly and embed zone, geometric model and finite element grid that this is regional are removed, the geometric model of assembly is embedded in the geometric model of supporting construction, the geometrical boundary of assembly and supporting construction of take is set up transition face as boundary, divides respectively quadrilateral and triangular finite element grid on device region and transition face.

Interference constraint between assembly and assembly, the filling layout optimization also needs Assurance component to be included in design section inside fully, document adopts the larger envelope circle of radius to carry out piecewise approximation to the Polygonal Boundary of design section, now contains to retrain can be similar to equally to adopt center of circle distance function to describe.N assembly even arranged, i assembly (i=1,2 ..., n) use N _iindividual envelope circle is approximate, and design domain is approximate by M envelope circle, retrains this n assembly and does not interfere and be included in design domain and need to have

individual equation of constraint.

Although the disclosed method of document can solve between restraint assembly, the interference problem on assembly and design domain border, has introduced a large amount of equation of constraint.If for example judgement has 4 assemblies, each assembly approximate with 4 envelopes circles, 4 approximate interference problems of envelope circle for design domain, the distance of center circle that need to judge altogether 160 two circles from radius and relative size (160 equation of constraint).So many equation of constraint number causes topology optimization problem to be difficult to carry out or is difficult to convergence.

Summary of the invention

In order to overcome existing method constraint, too much cause optimization problem to be difficult to carry out or be difficult to the deficiency of convergence, the invention provides a kind of method for layout optimal design of multi-assembly structure system based on the P norm.The method is used the P norm to interfere the equation of constraint cohesion for one or more equation of constraint all component, all interference constraints between assembly and between assembly and design domain are surrounded as the feasible zone of assembly interference problem, by the P norm, interfere equation of constraint to be condensed into one or several equation of constraint all assemblies, the maximal value that the value of the P norm obtained is approached all constraint functions, and be the conservative estimation to assembly interference problem feasible zone, make to optimize and can in the scope of former feasible zone, find part or globally optimal solution.The constraint condensing method that the present invention proposes is in conjunction with the assembly synergistic method of topological optimization design based on background grid and density points technology, solved in traditional assembly synergistic method of topological optimization design and can't carry out the problem that maybe can't restrain owing to interfering the constraint number too much to cause optimizing, interfere equation of constraint to be condensed into one or several equation of constraint all assemblies, reduced the number of interfering equation of constraint, made optimization problem carry out smoothly and to restrain.

The technical solution adopted for the present invention to solve the technical problems is: a kind of method for layout optimal design of multi-assembly structure system based on the P norm is characterized in comprising the following steps:

(a) set up finite element model by the cad model of structure, the multicomponent structures finite element model is divided into to structured grid, background grid and component grid three parts, definition load and boundary condition.

(b) by assembly and design domain boundary demarcation envelope circle, set up equation of constraint:

$\left\{\begin{array}{c}\∀i=\mathrm{1,2},...,n;j=i+1,...,n;\∀k=\mathrm{1,2},...,N_i;l=\mathrm{1,2},...,N_j\\ s.t.:C_{\mathrm{ij}}^{\mathrm{kl}}=\frac{\left|\right|O_{i\_k}{O}_{j\_l}\left|\right|}{R_{i\_k}+R_{j\_l}}\≥1\\ \∀\mathrm{\ϵ}=\mathrm{1,2},...,n;\∀\mathrm{\τ}=\mathrm{1,2},...,N_p;\mathrm{\ζ}=\mathrm{1,2},...,M\\ s.t.:C_{\mathrm{\ϵ}}^{\mathrm{\τ\ζ}}=\frac{\left|\right|O_{\mathrm{\ϵ}\_\mathrm{\τ}}{O}_{\mathrm{\ζ}}\left|\right|}{R_{\mathrm{\ϵ}\_\mathrm{\τ}}+R_{\mathrm{\ζ}}}\≥1\end{array}\right.$

Wherein, n is component count; N _ifor being used for being similar to the envelope circle number of i assembly; O _{i_k}, R _{i_k}be respectively the center of circle and the radius of k envelope circle of i assembly; The number of the envelope circle that M is the Approximate Design zone; be respectively the center of circle and radius for τ the large envelope circle in Approximate Design zone.

Above-mentioned equation of constraint is integrated with a P norm constraint equation:

$C_{\mathrm{PN}}={(\underset{i=1}{\overset{n-1}{\mathrm{\Σ}}}\underset{j=i+1}{\overset{n}{\mathrm{\Σ}}}\underset{k=1}{\overset{N_i}{\mathrm{\Σ}}}\underset{l=1}{\overset{N_j}{\mathrm{\Σ}}}{\left(C_{\mathrm{ij}}^{\mathrm{kl}}\right)}^p+\underset{\mathrm{\ϵ}=1}{\overset{n}{\mathrm{\Σ}}}\underset{\mathrm{\τ}=1}{\overset{N_{\mathrm{\ϵ}}}{\mathrm{\Σ}}}\underset{\mathrm{\ζ}=1}{\overset{M}{\mathrm{\Σ}}}{\left(C_{\mathrm{\ϵ}}^{\mathrm{\τ\ζ}}\right)}^p)}^{\frac{1}{p}}\≥1$

Wherein, p is the parameter of P norm.

(c) setting up Topological optimization model is:

Find η=(η ₁, η ₂..., η _enum); S=(s ₁, s ₂... s _n), s wherein _i=(x _i, y _i, θ _i)

min φ(η,S)

s.t. KU＝F

$G_j(\mathrm{\η},S)\≤\stackrel{\‾}{G_j},j=1,2,...,J$

C _PN≥1

Wherein, η is the pseudo-intensity vector in unit on design domain; Enum is design domain grid number; The Position Design variable that S is assembly, wherein s _i=(x _i, y _i, θ _i) represent respectively x coordinate, y coordinate and the direction coordinate of i assembly barycenter; N is component count; The objective function that φ (η, S) is topology optimization problem; K is finite element model global stiffness matrix; F is the node equivalent load vectors; U is node global displacement vector; G _j(η, S) is j constraint function;

it is the upper limit of j constraint function; J is the number of constraint; C _pNfor P norm constraint equation.

(d) model is carried out to a finite element analysis, respectively geometry variable and pseudo-density variables are carried out to sensitivity analysis, try to achieve the sensitivity of objective function and constraint condition, choose optimized algorithm GCMMA and be optimized design, result is optimized.

The invention has the beneficial effects as follows: the method is used the P norm to interfere the equation of constraint cohesion for one or more equation of constraint all component, all interference constraints between assembly and between assembly and design domain are surrounded as the feasible zone of assembly interference problem, by the P norm, interfere equation of constraint to be condensed into one or several equation of constraint all assemblies, the maximal value that the value of the P norm obtained is approached all constraint functions, and be the conservative estimation to assembly interference problem feasible zone, make to optimize and can in the scope of former feasible zone, find part or globally optimal solution.The constraint condensing method that this invention proposes is in conjunction with the assembly synergistic method of topological optimization design based on background grid and density points technology, solved in assembly synergistic method of topological optimization design traditional in the background technology and can't carry out the problem that maybe can't restrain owing to interfering the constraint number too much to cause optimizing, interfere equation of constraint to be condensed into one or several equation of constraint all assemblies, reduced the number of interfering equation of constraint, made optimization problem carry out smoothly and to restrain.

Below in conjunction with drawings and Examples, the present invention is elaborated.

The accompanying drawing explanation

Fig. 1 is the structural representation of multicomponent structures system layout optimal design in background technology.

Fig. 2 is the schematic diagram that in background technology, the limited envelope circle of assembly and design section border is divided.

Fig. 3 be in background technology in multicomponent structures grid divide schematic diagram.

Fig. 4 is moulded dimension and the Boundary Conditions in Structures schematic diagram of specific embodiment.

Fig. 5 is that the limited envelope circle of specific embodiment is divided schematic diagram.

Fig. 6 is the Cooperative Optimization result of specific embodiment application the inventive method.

Embodiment

With reference to Fig. 4-6.The multicompartment cantilever beam structure under fixed load of take is example explanation the present invention.The plane cantilever beam structure is of a size of long 90mm, high 30mm, and its Young modulus is 7 * 10 ¹⁰mpa, Poisson ratio is 0.3.Four assemblies of the inner embedding of cantilever beam structure, be respectively rectangular module, convex shape assembly, cruciform component and L shaped assembly.Initial center position and the inceptive direction of rectangular module is (15,15,360); Initial center position and the inceptive direction of convex shape assembly is (35,15,360); Initial center position and the inceptive direction of cruciform component is (55,15,360); Center initial position and the inceptive direction of L shaped assembly is (75,15,360).The Young modulus of each assembly is 2 * 10 ¹¹mpa, Poisson ratio is 0.3.Design semi-girder load-carrying construction and the position of assembly in load-carrying construction, make its rigidity maximum, and overall material usage volume fraction is 50% to the maximum.The concrete grammar step is as follows:

(a) finite element modeling.

Cad model by structure is set up finite element model: setting the grid length of side is 1mm, and the finite element model of multicompartment cantilever beam structure is divided into to structured grid, component grid, three parts of background grid.Definition boundary condition: right-hand member summit, cantilever beam structure coboundary and coboundary are fixed apart from right endpoint 30mm place node; On cantilever beam structure lower boundary left end summit and lower boundary, apart from left end point formula place node, apply along the load of x axle negative sense and y axle negative sense, magnitude of load is 100000N.

(b) by assembly and design domain boundary demarcation envelope circle, set up equation of constraint.

By rectangular module, convex shape assembly, cruciform component and L shaped assembly border, respectively with 2,3,1 and 3 next being similar to of envelope circle, by the design domain border, with 4 envelopes circles, come approximate.The method of application reference document, set up interference equation of constraint between assembly and the containing equation of constraint between assembly and design domain border, has 127 equation of constraint:

$\left\{\begin{array}{c}\∀i=\mathrm{1,2,3,4};j=i+1,...,4;\∀k=\mathrm{1,2},...,N_i;l=\mathrm{1,2},...,N_j\\ s.t.:C_{\mathrm{ij}}^{\mathrm{kl}}=\frac{\left|\right|O_{i\_k}{O}_{j\_l}\left|\right|}{R_{i\_k}+R_{j\_l}}\≥1\\ \∀\mathrm{\ϵ}=\mathrm{1,2,3,4};\∀\mathrm{\τ}=\mathrm{1,2},...,N_{\mathrm{\ϵ}};\mathrm{\ζ}=\mathrm{1,2,3,4}\\ s.t.:C_{\mathrm{\ϵ}}^{\mathrm{\τ\ζ}}=\frac{\left|\right|O_{\mathrm{\ϵ}\_\mathrm{\τ}}{O}_{\mathrm{\ζ}}\left|\right|}{R_{\mathrm{\ϵ}\_\mathrm{\τ}}+R_{\mathrm{\ζ}}}\≥1\end{array}\right.$

O wherein _{i_k}, R _{i_k}be respectively the center of circle and the radius of k envelope circle of i assembly;

,

be respectively the center of circle and radius for τ the large envelope circle in Approximate Design zone;

be that interference between l envelope of k envelope circle of i assembly and j assembly justified retrains;

be the of τ envelope circle of ε assembly and design domain border

containing constraint between individual envelope circle.

Above-mentioned 127 equation of constraint are integrated into to an equation of constraint by the P norm:

$C_{\mathrm{PN}}={(\underset{i=1}{\overset{3}{\mathrm{\Σ}}}\underset{j=i+1}{\overset{4}{\mathrm{\Σ}}}\underset{k=1}{\overset{4}{\mathrm{\Σ}}}\underset{l=1}{\overset{4}{\mathrm{\Σ}}}{\left(C_{\mathrm{ij}}^{\mathrm{kl}}\right)}^p+\underset{\mathrm{\ϵ}=1}{\overset{4}{\mathrm{\Σ}}}\underset{\mathrm{\τ}=1}{\overset{4}{\mathrm{\Σ}}}\underset{\mathrm{\ζ}=1}{\overset{4}{\mathrm{\Σ}}}{\left(C_{\mathrm{\ϵ}}^{\mathrm{\τ\ζ}}\right)}^p)}^{\frac{1}{p}}\≥1$

(c) setting up Topological optimization model is:

Find η=(η ₁, η ₂..., η _enum); S=(s ₁, s ₂... s _n), s wherein _i=(x _i, y _i, θ _i)

$\begin{array}{cc}\min & \mathrm{\φ}(\mathrm{\η},S,U)=\mathrm{UKU}=\underset{e=1}{\overset{\mathrm{enum}}{\mathrm{\Σ}}}{u}_e{k}_e{u}_e\end{array}$

s.t.KU＝F

$V(\mathrm{\η},S)\≤\stackrel{\‾}{V}$

C _PN≥1

Wherein, η is the pseudo-intensity vector in unit on design domain; Enum is design domain grid number; The Position Design variable that S is assembly, wherein s _i=(x _i, y _i, θ _i) represent respectively x coordinate, y coordinate and the direction coordinate of i assembly barycenter; N is component count;

for the objective function of topology optimization problem, in this problem, be the structure compliance, be numerically equal to the bulk strain energy of structure; K is finite element model global stiffness matrix; F is the node equivalent load vectors; U is node global displacement vector; V (η, S) is the volume constraint function; for the upper limit of volume constraint,

v wherein ₀for the volume of material cell is arranged; C _pNfor P norm constraint equation.

(d) finite element analysis and Optimization Solution.

By finite element soft Ansys, model is carried out to a finite element analysis; Be optimized sensitivity analysis by structure optimization platform Boss-Quattro again, try to achieve the sensitivity of objective function and constraint function, choose gradient optimal method GCMMA(Globally Convergent Method of Moving Asymptotes) optimized algorithm is optimized design, and result is optimized.

As seen from Figure 6, by the inventive method, carry out multicomponent structures system layout optimal design, a large amount of equation of constraint can be condensed into to one or more P norm constraint equations, and can guarantee that the equation of constraint after cohesion meets institute's Constrained.With the method in list of references, compare, method used in the present invention can solve because of component count and envelope circle number and too much introduce a large amount of equation of constraint, causes optimization problem to be difficult to carry out or be difficult to the problem restrained.Therefore, the method applied in the present invention applicability is wider.

Claims (1)

1. the method for layout optimal design of multi-assembly structure system based on the P norm is characterized in that comprising the following steps:

(a) set up finite element model by the cad model of structure, the multicomponent structures finite element model is divided into to structured grid, background grid and component grid three parts, definition load and boundary condition;

(b) by assembly and design domain boundary demarcation envelope circle, set up equation of constraint:

$\left\{\begin{array}{c}\∀i=\mathrm{1,2},...,n;j=i+1,...,n;\∀k=\mathrm{1,2},...,N_i;l=\mathrm{1,2},...,N_j\\ s.t.:C_{\mathrm{ij}}^{\mathrm{kl}}=\frac{\left|\right|O_{i\_k}{O}_{j\_l}\left|\right|}{R_{i\_k}+R_{j\_l}}\≥1\\ \∀\mathrm{\ϵ}=\mathrm{1,2},...,n;\∀\mathrm{\τ}=\mathrm{1,2},...,N_p;\mathrm{\ζ}=\mathrm{1,2},...,M\\ s.t.:C_{\mathrm{\ϵ}}^{\mathrm{\τ\ζ}}=\frac{\left|\right|O_{\mathrm{\ϵ}\_\mathrm{\τ}}{O}_{\mathrm{\ζ}}\left|\right|}{R_{\mathrm{\ϵ}\_\mathrm{\τ}}+R_{\mathrm{\ζ}}}\≥1\end{array}\right.$

Wherein, n is component count; N _ifor being used for being similar to the envelope circle number of i assembly; O _{i_k}, R _{i_k}be respectively the center of circle and the radius of k envelope circle of i assembly; The number of the envelope circle that M is the Approximate Design zone;

be respectively the center of circle and radius for τ the large envelope circle in Approximate Design zone;

Above-mentioned equation of constraint is integrated with a P norm constraint equation:

$C_{\mathrm{PN}}={(\underset{i=1}{\overset{n-1}{\mathrm{\Σ}}}\underset{j=i+1}{\overset{n}{\mathrm{\Σ}}}\underset{k=1}{\overset{N_i}{\mathrm{\Σ}}}\underset{l=1}{\overset{N_j}{\mathrm{\Σ}}}{\left(C_{\mathrm{ij}}^{\mathrm{kl}}\right)}^p+\underset{\mathrm{\ϵ}=1}{\overset{n}{\mathrm{\Σ}}}\underset{\mathrm{\τ}=1}{\overset{N_{\mathrm{\ϵ}}}{\mathrm{\Σ}}}\underset{\mathrm{\ζ}=1}{\overset{M}{\mathrm{\Σ}}}{\left(C_{\mathrm{\ϵ}}^{\mathrm{\τ\ζ}}\right)}^p)}^{\frac{1}{p}}\≥1$

Wherein, p is the parameter of P norm;

(c) setting up Topological optimization model is:

Find η=(η ₁, η ₂..., η _enum); S=(s ₁, s ₂... s _n), s wherein _i=(x _i, y _i, θ _i)

min φ(η,S)

s.t. KU＝F

$G_j(\mathrm{\η},S)\≤\stackrel{\‾}{G_j},j=\mathrm{1,2}...,J$

C _PN≥1

Wherein, η is the pseudo-intensity vector in unit on design domain; Enum is design domain grid number; The Position Design variable that S is assembly, wherein s _i=(x _i, y _i, θ _i) represent respectively x coordinate, y coordinate and the direction coordinate of i assembly barycenter; N is component count; The objective function that φ (η, S) is topology optimization problem; K is finite element model global stiffness matrix; F is the node equivalent load vectors; U is node global displacement vector; G _j(η, S) is j constraint function;

it is the upper limit of j constraint function; J is the number of constraint; C _pNfor P norm constraint equation;

(d) model is carried out to a finite element analysis, respectively geometry variable and pseudo-density variables are carried out to sensitivity analysis, try to achieve the sensitivity of objective function and constraint condition, choose optimized algorithm GCMMA and be optimized design, result is optimized.

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