CN106844897A - A kind of tree topology wound structure method based on OPTIMIZATION OF CONTINUUM STRUCTURES - Google Patents

Abstract

The invention discloses a kind of tree topology wound structure method based on OPTIMIZATION OF CONTINUUM STRUCTURES, the tree topology wound structure method based on OPTIMIZATION OF CONTINUUM STRUCTURES, using OPTIMIZATION OF CONTINUUM STRUCTURES ICM methods, the topological optimization model of structure total strain energy minimization under weight constraints is set up, Optimized Iterative form is set up according to the saddle point condition of single constrained optimization problem.The present invention need not be previously set the information such as the number of plies, floor height, branch amount, roof structure and the tree support connection position of tree, with bigger design space.The numerical example shows that the wound structure of the tree based on structural Topology Optimization ICM methods for being proposed is feasible, can provide diversified topographic morphologies for the tree of conceptual phase.

Description

A kind of tree topology wound structure method based on OPTIMIZATION OF CONTINUUM STRUCTURES

Technical field

The invention belongs to build the topology wound structure technical field of the tree in large space supporting structure form, it is proposed that A kind of tree topology wound structure new method based on OPTIMIZATION OF CONTINUUM STRUCTURES.

Background technology

Tree is proposed by German's Otto (Frei Otto) first the sixties in 20th century, is according in nature The bionical building structure type of ecology and the Force principle design of tree.The Stuttgart, Germany airport built up for 1991 is Otto The tree typical engineering examples of design, three layers of branch tree sky wide for terminal is provided of its supporting roof system Between.Tree has Path of Force Transfer rationally, and Support cover scope is wide, can form larger supporting and space etc. with less rod member Advantage, is more and more applied in airport, station, large public activity center etc. are built, and such as China Railway High-speed is long Husky southern station, Qatar National Convention Center and Bombay Tote hotels etc..Tree rod member is more, form complicated, and form is created is The problem of preferential solution is needed in its structure design.On the premise of building function demand is met, design trunk height, branch layer Support Position of number, bifurcation site and number and roof structure etc., makes each component arrangement meet optimal Path of Force Transfer, embodies power Stream realizes the perfect adaptation of shape and power from top to bottom, from rudimentary branch to the convergence process of senior branch.Common tree Morphological research method has three classes：Test method, method of geometry and numerical method.Common test method have wet thread model method, Shredded dried bean curd line model method and silk thread beading modelling etc..Method of geometry application fractal theory carries out geometry wound structure to tree. Gawell describes the tree fractal Morphology based on the generation of L- systems and Bombay Tote hotels based on the method design.With The development of numerical analysis on structure and Optimization Design, using the research that numerical method is created to the form of tree And application turns into focus.Von Buelow applications genetic algorithm finds the shortest path of tree component, proposes a kind of tree-shaped Structure looks for shape method.Tree is assumed to full articulation model by Hunt etc., the void that then addition can be vertically slidable on model Bearing, by adjusting internal node coordinate up to empty end reaction is zero, finally determines planform.Wu Yue etc. proposes tree-shaped knot The inverse of structure hangs recursion and looks for shape method.Open pretty grade has carried out looking for shape to study based on continuous broken line cable elements to tree.Using skeleton Structural optimization method, Cui Changyu etc. propose the tree form wound structure method based on susceptibility, meanwhile, improve Continuum Structure The evolutional structure optimization (ESO) of topological optimization carries out topology wound structure to tree structure.The ESO structures of Sasaki application enhancements are opened up Flutter optimization method and devise Qatar National Convention Center.

But in current institute's application test method, 3 kinds of methods of method of geometry and numerical method, test method is due to by model ratio The influence of example chi is larger, using being restricted；The tree of fractal method generation has only focused on the morphological feature of tree, its geometry Configuration is needed by the further optimization design of structure optimization technology to consider its mechanical characteristics；Topology employed in numerical method Optimization ESO methods have that Optimized Iterative number of times is too many, solution efficiency is low, the unstable key of algorithm, often resulted in not using different deletion rates The deficiencies such as same optimum structure.And preceding 2 kinds of methods need to be manually set the number of plies of tree, floor height, branch amount, roofing knot The information such as structure and tree support connection position, design space is limited.

The content of the invention

OPTIMIZATION OF CONTINUUM STRUCTURES is commonly available the optimal topographic morphologies of matrix morphology, it is not necessary to artificial to specify tree The condition of some priori such as the floor height of shape structure, the number of plies and number of branches, with bigger design space.The purpose of the present invention exists Structure method is created in a kind of tree topology based on OPTIMIZATION OF CONTINUUM STRUCTURES is provided, it is intended to which what solution was applied at present opens up Optimization ESO methods are flutterred to have that Optimized Iterative number of times is too many, solution efficiency is low, the unstable key of algorithm, often resulted in using different deletion rates The problem of different optimum structure, it is to avoid other tree-shaped wound structure methods for designing need to pre-set trunk height, the branch number of plies, divide Vent put and number and roof structure the deficiency such as Support Position.

Tree topology wound structure method based on OPTIMIZATION OF CONTINUUM STRUCTURES of the present invention realizes that technology is：Using OPTIMIZATION OF CONTINUUM STRUCTURES ICM methods, set up the topological optimization model of structure total strain energy minimization under weight constraints；According to single The saddle point condition of constrained optimization problem establishes Optimized Iterative form.

It is described based on OPTIMIZATION OF CONTINUUM STRUCTURES tree topology wound structure method topological optimization model be：

Model is set up with ICM methods, topological variable is expanded to the continuous variable on 0 to 1 interval, each list by discrete variable First weight and element stiffness battle array are identified with different filter functions respectively from the relation between topology design variable：

Wherein w_iAnd k_iBe unit weight and element stiffness battle array,AndIt is the intrinsic weight of unit and unit self-stiffness battle array； Filter function is taken as：

Wherein, power exponent α_wAnd α_k1 and 3 can respectively be taken；

Then unit strain energy is expressed as：

WhereinTopological variate-value during for kth time iteration,The strain energy of unit i during kth time iteration；Unit weight It is expressed as：

Thus obtain with ICM methods set up Optimized model be：

For when preventing topological value from taking 0 structural stiffness matrix be likely to occur it is unusual, it is usual with a small valuet_i Instead of 0, can uset_i = 0.01；

Described method for solving is：

Because formula (5) is single restricted problem, the constraint must take equation, meaningless otherwise as unconstrained problem, noteDefinition actively collects I_a=i |t_i ＜ t_i＜ 1 (i=1 ..., N) } then formula (5) be：

The Augmented Lagrangian Functions of the problem are：

The saddle point condition that above formula takes extreme value is：

Thus：

To be obtained in equality constraint in formula (9) substitution formula (6)：

So as to obtain：

Formula (12) substitutes into formula (10) and obtains：

In view of the Operations of Interva Constraint of topological variable, that is,：

Update and actively collect, return again to calculate t by formula (12)_i, so circulate, until actively collecting constant terminates circulation, try to achieve Optimal solution t^*Also the optimal solution of formula (5) is, structure is changed by formula (1), into circulating next time, such iteration is received until meeting Hold back criterion：

Δ e=| (e^(k+1)-e^(k))/e^(k+1)|≤ε (14)

Wherein, e^(k)And e^(k+1)Be the structure total strain energy of front-wheel and epicycle iteration, ε is convergence precision, take herein ε= 0.001。

It is a kind of using the tree based on OPTIMIZATION OF CONTINUUM STRUCTURES another object of the present invention is to provide Topology wound structure method provides tree-shaped supporting structure in building hall.

It is a kind of using the tree based on OPTIMIZATION OF CONTINUUM STRUCTURES another object of the present invention is to provide Topology wound structure method provides tree-shaped supporting structure in airport building.

It is a kind of using the tree based on OPTIMIZATION OF CONTINUUM STRUCTURES another object of the present invention is to provide Topology wound structure method provides tree-shaped supporting structure in high-speed railway waiting hall.

It is a kind of using the tree based on OPTIMIZATION OF CONTINUUM STRUCTURES another object of the present invention is to provide Topology wound structure method provides tree-shaped supporting structure in large public activity center.

The topology wound structure method of the tree based on OPTIMIZATION OF CONTINUUM STRUCTURES that the present invention is provided, structural Topology Optimization Independent Continuous Mappings (ICM) method sets up Optimized model, is solved using dual sequence quadratic programming, and Optimization Solution efficiency is higher, Optimized Iterative number of times is generally between 30~50 times, and the Optimized Iterative number of times that the technology of present application ESO methods needs leads to Often more than 100 times.Present invention application (ICM) method sets up the topology wound structure method of tree；For conceptual phase tree Shape structural topology creates structure problem, proposes that carrying out wound structure using OPTIMIZATION OF CONTINUUM STRUCTURES method designs；Using Continuum Structure Topological optimization ICM methods, set up the topological excellent of structure total strain energy minimization under weight constraints (i.e. structure global stiffness maximization) Change model, Optimized Iterative form is established according to the saddle point condition of single constrained optimization problem；With the wound structure of a plane tree As a example by, weight ratio, roof structure rigidity, roof structure geometric format, design space height etc. are discussed to tree topology The influence of form, and give the example of several application scenarios；The numerical example shows that proposed tree creates the structure of structure Topology Optimization Method is feasible, can provide diversified topographic morphologies for conceptual phase tree.

Brief description of the drawings

Fig. 1 is the tree topology wound structure method stream based on OPTIMIZATION OF CONTINUUM STRUCTURES provided in an embodiment of the present invention Cheng Tu.

Fig. 2 is plane tree design condition schematic diagram provided in an embodiment of the present invention.

Fig. 3 is different volumes provided in an embodiment of the present invention than lower tree topological structure schematic diagram；

In figure：A () weight compares 5%；B () weight compares 10%；C () weight compares 20%.

Fig. 4 is tree topology schematic diagram under roof structure different-stiffness provided in an embodiment of the present invention；

In figure：(a) E=2.1 × 10⁵Mpa；(b) E=2.1 × 10⁶MPa；(c) E=2.1 × 10⁷MPa；(d) E=2.1 × 10⁸Mpa。

Fig. 5 is tree topological structure schematic diagram under design section different height provided in an embodiment of the present invention；

In figure：(a)5m；(b)7.5m；(c)10m；(d)12.5m；(e)15m；(f)20m.

Fig. 6 is tree topological structure schematic diagram under roofing different shape provided in an embodiment of the present invention；

In figure：(a) inclined slope roof；(b) circular arc roofing.

Fig. 7 is wall load-bearing skeleton tree topology wound structure structural representation provided in an embodiment of the present invention；

In figure：(a) finite element grid；(b) tree topology.

Fig. 8 is three-dimensional tree topology wound structure structural representation provided in an embodiment of the present invention；

In figure：(a) finite element grid；(b) tree topology.

Fig. 9 is many tree topology wound structure structural representations of plane provided in an embodiment of the present invention；

In figure：Many tree topology wound structure design conditions of (a) plane；Many tree topology forms of (b) plane.

Specific embodiment

In order to make the purpose , technical scheme and advantage of the present invention be clearer, with reference to embodiments, to the present invention It is further elaborated.It should be appreciated that the specific embodiments described herein are merely illustrative of the present invention, it is not used to Limit the present invention.

Application principle of the invention is explained in detail below in conjunction with the accompanying drawings.

As shown in figure 1, the tree topology wound structure based on OPTIMIZATION OF CONTINUUM STRUCTURES provided in an embodiment of the present invention Method is comprised the following steps：

S101：Using OPTIMIZATION OF CONTINUUM STRUCTURES ICM methods, under setting up weight constraints, structure total strain energy minimization is (i.e. Structure global stiffness maximize) topological optimization model；

S102：Optimized Iterative form is established according to the saddle point condition of single constrained optimization problem.

Application principle of the invention is further described with reference to specific embodiment.

1. tree creates the topology optimization of structure

1.1 trees create the topological optimization model of structure

Tree topology wound structure applies the conceptual phase in structure design, is generally only considering vertical roof load Under effect, the tree topographic morphologies of maximizing stiffness are designed.Thus, it is excellent that it can be expressed as a Continuum Structure topology Change problem, i.e.,：Under specified material usage, in specified design section, topological optimization tree form makes specifying Roof load effect under, the global stiffness of structure is very big.Global stiffness maximization and the total strain energy pole of structure due to structure Smallization is equivalent, so, the topological optimization model of tree wound structure is attributed to：The lower total strain energy of weight (or volume) constraint The topology optimization problem of minimization, as shown in formula (1).

Wherein, t_iIt is topology design variable, e is structure total strain energy, and W is structure gross weight.

The modeling and solution of total strain energy minimization problem under 1.2 weight constraints

Under constraining constant weight, the minimum OPTIMIZATION OF CONTINUUM STRUCTURES problem for turning to target of structure total strain energy is used ICM methods set up model, and topological variable is expanded to the continuous variable on 0 to 1 interval, each unit weight and list by discrete variable Relation between first Stiffness Matrix and topology design variable is identified with different filter functions respectively：

Wherein w_iAnd k_iBe unit weight and element stiffness battle array,AndIt is the intrinsic weight of unit and unit self-stiffness battle array. Filter function is taken as：

Wherein, power exponent α_wAnd α_k1 and 3 can respectively be taken.

Then unit strain energy can be expressed as：

WhereinTopological variate-value during for kth time iteration,The strain energy of unit i during kth time iteration.

Unit weight can be expressed as：

Thus obtain with ICM methods set up Optimized model be：

For when preventing topological value from taking 0 structural stiffness matrix be likely to occur it is unusual, it is usual with a small valuet_i Instead of 0, can uset_i = 0.01。

Because formula (4) is single restricted problem, the constraint must take equation, meaningless otherwise as unconstrained problem, noteDefinition actively collects I_a=i |t_i ＜ t_i＜ 1 (i=1 ..., N) } then formula (6) be：

The Augmented Lagrangian Functions of the problem are：

The saddle point condition that above formula takes extreme value is：

Thus：

To be obtained in equality constraint in formula (10) substitution formula (7)：

So as to obtain：

Formula (12) substitutes into formula (10) and obtains：

In view of the Operations of Interva Constraint of topological variable, that is,：

Update and actively collect, return again to calculate t by formula (13)_i, so circulate, until actively collecting constant terminates circulation, try to achieve Optimal solution t^*Also the optimal solution of formula (6) is, structure is changed by formula (2), into circulating next time, such iteration is received until meeting Hold back criterion：

Δ e=| (e^(k+1)-e^(k))/e^(k+1)|≤ε (15)

Wherein, e^(k)And e^(k+1)Be the structure total strain energy of front-wheel and epicycle iteration, ε is convergence precision, take herein ε= 0.001。

2. tree topology creates structure

Example 1：Plane tree topographic morphologies create structure

As shown in Fig. 2 design section is 10m × 7.5m rectangular areas, top is acted on q=1kN/m by vertical uniform load, Load action side 0.3m thickness, as roof structure, is Non-design region.It is tree in the 0.4m regions of bottom middle part Root FX.Material is steel, elastic modulus E=2.1 × 10⁵MPa, Poisson's ratio 0.3.It is for about with certain weight ratio Beam, minimization structure total strain energy obtains the tree topographic morphologies of rigidity of structure maximization.

Fig. 3 shown in the case where roof structure rigidity is constant, Different Weight ratio (reserved materials weight/initial designs Region material gross weight) the lower tree topographic morphologies change of constraint.When material usage is different, the topographic morphologies of tree It is different, with the increase of material usage, tree topology becomes increasingly complex.

In the case of Fig. 4 shows that certain weight ratio constraint (10%) is constant, tree during roof structure stiffness change Topographic morphologies change.Herein for model treatment is convenient, simulated by setting the dual extension-compression modulus of roofing Rotating fields different firm Degree, rather than roofing layer geometrical scale is changed, data shown in Fig. 4 are the dual extension-compression modulus value for setting.Can from Fig. 4 To see, with the increase of the roofing layer rigidity of structure, the bifurcated of tree is fewer and feweri, or even deteriorates to a root post.

Fig. 5 shows roof structure rigidity (E=2.1 × 10⁵MPa) and in the case of weight is constant than constraint (10%), Tree topographic morphologies change when changing the height of design section.When highly smaller (Fig. 5 a~c), main trunk will not go out Existing, at this moment, design section height is different, and the branch topographic morphologies of tree are also different.When height than it is larger when (Fig. 5 c~Fig. 5 f), it is main Trunk occurs, and increases the topographic morphologies that design section highly, not changes tree, and simply main trunk height increases.

Influence to tree topology form when Fig. 6 shows that roof structure geometric shape changes, Fig. 6 a are face of slope shape Formula, its tree topology is entirely different with horizental roof, and sturdy trunk is inclined to side high.Fig. 6 b are circular roofing shape Formula, the bifurcated of branch is no longer y-bend form, and occurs in that the form of trifid structure.

Example 2：Wall load-bearing skeleton tree topology creates structure

Using tree as the skeleton of wall, the load of wall can be efficiently transmitted, can reach again and preferably regard Feel effect.The tree-shaped skeleton structure form wound structure of plane can be that the wall of this form brings diversity, be the hair of epidermal structure Exhibition provides a kind of new approaches.Chongqing Jiang Wancheng is case history of the tree as wall bone bearing frame.

As shown in Fig. 7 left figures, design section is 10m × 10m squares four sides wall, and vertical uniform lotus is received on 8m high, top Load acts on q=1kN/m, and load action side 0.2m thickness, as wall girder construction, is Non-design region.The angle point of bottom 4 Region 0.2m wide is the root FX of tree.Material is steel, and material parameter is same as Example 1.Take weight ratio about Beam is 10%, minimize structure total strain energy, the tree topographic morphologies that topological optimization is obtained as shown in Fig. 7 right figures, each Angle stretches out two sides tree structure on the face for intersecting vertically, and in the same face, two tree structures stretch to centre, intersects near Seemingly for the arch structure form commonly used in an arch, with engineering is perfectly in harmony.

Example 3：Three-dimensional tree topology creates structure

As shown in Fig. 8 left figures, design section is the cubical area of 20m × 20m × 20m, and vertical uniform load is received on top Effect q=1kN/m², load action face 0.5m thickness, as roof structure, is Non-design region.Bottom zone line 1m × 1m It is the root FX of tree in region.Material is steel, and material parameter is identical with example 1.Weight ratio is taken to be constrained to 10%, structure total strain energy is minimized, the tree topographic morphologies that topological optimization is obtained are four bifurcateds as shown in Fig. 8 right figures Form, two-layer branch, but one layer above of branch branch of the height than following one layer is highly much smaller.

Example 4：Many tree topology wound structures of plane

As illustrated in fig. 9, design section is 50m × 7.5m rectangles, and top is acted on q=1kN/m, lotus by vertical uniform load Action edge 0.3m thickness is carried as roof structure, is Non-design region.Bottom is divided into 4 across in 5 points of supports, each support area 0.3m wide, as the root FX of tree.Material is steel, and material parameter is same as Example 1.It is than 10% with weight Constraint, minimization structure total strain energy obtains the tree topographic morphologies of rigidity of structure maximization as shown in figure 9b, due to not Deng across resulting tree is asymmetric form, and equally, adjacent tree stretches and intersects approximate arch.

The numerical example of the invention shows that it is feasible to carry out tree wound structure using OPTIMIZATION OF CONTINUUM STRUCTURES technology , can provide diversified selection scheme for the topographic morphologies of tree in conceptual phase.Different from fractal method The topology of spanning tree does not account for the loading characteristic of tree, is needed also different from some methods for being based on mechanical model artificial Some parameters such as floor height, the number of plies, the bifurcated number of tree are specified, thus optimal tree can be sought in bigger design space Shape structural topology form.The rigidity of roof structure has a significant impact to the topographic morphologies of tree, thus, in topology design When, roof structure should participate in mechanics optimization design together with design section, and answer accurate simulation its rigidity.Different weight ratios are about Beam has a significant impact to the topographic morphologies of tree, sets an appropriate weight ratio so that the rigidity and room of tree When the rigidity of face structure is close, more satisfactory tree topology form can be obtained.Design section and roofing geometric format are to tree Topographic morphologies are had a significant impact, and parameter is clearly specified and accurate simulation preferably in conceptual design.

Presently preferred embodiments of the present invention is the foregoing is only, is not intended to limit the invention, it is all in essence of the invention Any modification, equivalent and improvement made within god and principle etc., should be included within the scope of the present invention.

Claims (7)

1. a kind of tree topology based on OPTIMIZATION OF CONTINUUM STRUCTURES creates structure method, it is characterised in that described based on even The tree topology wound structure method application OPTIMIZATION OF CONTINUUM STRUCTURES ICM methods of continuous body structural Topology Optimization, set up weight about The topological optimization model of structure total strain energy minimization under beam；

Under constraining constant weight, the minimum OPTIMIZATION OF CONTINUUM STRUCTURES problem for turning to target of structure total strain energy uses ICM side Method sets up model, and topological variable is expanded to the continuous variable on 0 to 1 interval by discrete variable, and each unit weight and unit are firm Degree battle array is identified with different filter functions respectively from the relation between topology design variable：

$w_i=f_w\left(t_i\right)w_i^0,k_i=f_k\left(t_i\right)k_i^0---\left(1\right)$

Wherein w_iAnd k_iBe unit weight and element stiffness battle array,AndIt is the intrinsic weight of unit and unit self-stiffness battle array；Filtering Function is taken as：

$f_w\left(t_i\right)=t^{{\mathrm{\α}}_w},f_k\left(t_i\right)=t^{{\mathrm{\α}}_k}---\left(2\right)$

Wherein, power exponent α_wAnd α_k1 and 3 can respectively be taken；

Then unit strain energy can be expressed as：

$e_i=\frac{1}{2}{u}_i^T{k}_i{u}_i=\frac{{\left(t_i^{{\mathrm{\α}}_k}\right)}^{\left(k\right)}}{2t_i^{{\mathrm{\α}}_k}}{u}_i^T{k}_i{u}_i=\frac{{\left(t_i^{{\mathrm{\α}}_k}\right)}^{\left(k\right)}}{t_i^{{\mathrm{\α}}_k}}{e}_i^{\left(k\right)}---\left(3\right)$

WhereinTopological variate-value during for kth time iteration,The strain energy of unit i during kth time iteration；

Unit weight can be expressed as：

$w_i=t_i^{{\mathrm{\α}}_w}{w}_i^0---\left(4\right)$

Thus the tree topology based on OPTIMIZATION OF CONTINUUM STRUCTURES for obtaining being set up with ICM methods creates structure method Topological optimization model is：

$\left\{\begin{array}{cc}Find& t\∈E^N\\ Make& \underset{i=1}{\overset{N}{\mathrm{\Σ}}}{\left(t_i^{{\mathrm{\α}}_k}\right)}^{\left(k\right)}{e}_i^{\left(k\right)}/t_i^{{\mathrm{\α}}_k}\→min\\ s.t.& \underset{i=1}{\overset{N}{\mathrm{\Σ}}}{t}_i^{{\mathrm{\α}}_w}{w}_i^0\≤\stackrel{\‾}{W}\\ & 0\≤t_i\≤1\end{array}\right.---\left(5\right)$

Wherein, t_iIt is topology design variable, e is structure total strain energy, and W is structure gross weight；w_iIt is unit weight and element stiffness Battle array,It is the intrinsic weight of unit and unit self-stiffness battle array；Power exponent α_w、α_k；WhereinTopology becomes during for kth time iteration Value.

2. the tree topology based on OPTIMIZATION OF CONTINUUM STRUCTURES as claimed in claim 1 creates structure method, and its feature exists In the saddle point condition of the single constrained optimization problem of foundation sets up Optimized Iterative form；The topological optimization model method for solving includes：

Because formula (5) is single restricted problem, the constraint must take equation, meaningless otherwise as unconstrained problem, noteDefinition actively collects I_a=i |t_i ＜ t_i＜ 1 (i=1 ..., N) } then formula (5) be：

$\left\{\begin{array}{cc}Find& t\∈E^N\\ Make& \underset{i\∈I_a}{\mathrm{\Σ}}{A}_i/t_i^{{\mathrm{\α}}_k}+\underset{i\∉I_a}{\mathrm{\Σ}}{A}_i/t_i^{{\mathrm{\α}}_k}\→min\\ s.t.& \underset{i\∈I_a}{\mathrm{\Σ}}{t}_i^{{\mathrm{\α}}_w}{w}_i^0+\underset{i\∉I_a}{\mathrm{\Σ}}{t}_i^{{\mathrm{\α}}_w}{w}_i^0=\stackrel{\‾}{W}\\ & \underset{\‾}{t_i}\≤t_i\≤1\end{array}\right.---\left(6\right)$

The Augmented Lagrangian Functions of the problem are：

$\begin{array}{c}L(t,\mathrm{\λ})=\underset{i\∈I_a}{\mathrm{\Σ}}{A}_i/t_i^{{\mathrm{\α}}_k}+\underset{i\∉I_a}{\mathrm{\Σ}}{A}_i/t_i^{{\mathrm{\α}}_k}\\ +\mathrm{\λ}(\underset{i\∈I_a}{\mathrm{\Σ}}{t}_i^{{\mathrm{\α}}_w}{w}_i^0+\underset{i\∉I_a}{\mathrm{\Σ}}{t}_i^{{\mathrm{\α}}_w}{w}_i^0-\stackrel{\‾}{W})\end{array}---\left(7\right)$

The saddle point condition that above formula takes extreme value is：

$\∂L/\∂t_i=-{\mathrm{\α}}_k{A}_i/t_i^{{\mathrm{\α}}_k+1}+{\mathrm{\α}}_w{\mathrm{\λw}}_i^0{t}_i^{{\mathrm{\α}}_w-1}=0---\left(8\right)$

Thus：

$t_i={\[{\mathrm{\α}}_k{A}_i/\left({\mathrm{\α}}_w{w}_i^0\right)\]}^{1/({\mathrm{\α}}_k+{\mathrm{\α}}_w)}{(1/\mathrm{\λ})}^{1/({\mathrm{\α}}_k+{\mathrm{\α}}_w)}---\left(9\right)$

To be obtained in equality constraint in formula (9) substitution formula (6)：

$\begin{array}{c}\underset{i\∈I_a}{\mathrm{\Σ}}{w}_i^0{\[{\mathrm{\α}}_k{A}_i/\left({\mathrm{\α}}_w{w}_i^0\right)\]}^{1/({\mathrm{\α}}_k+{\mathrm{\α}}_w)}{(1/\mathrm{\λ})}^{1/({\mathrm{\α}}_k+{\mathrm{\α}}_w)}\\ =\stackrel{\‾}{W}-\underset{i\∉I_a}{\mathrm{\Σ}}{t}_i^{{\mathrm{\α}}_w}{w}_i^0\end{array}---\left(10\right)$

So as to obtain：

${(1/\mathrm{\λ})}^{1/({\mathrm{\α}}_k+{\mathrm{\α}}_w)}=\frac{(\stackrel{\‾}{W}-\underset{i\∉I_a}{\mathrm{\Σ}}{t}_i^{{\mathrm{\α}}_w}{w}_i^0)}{\underset{i\∈I_a}{\mathrm{\Σ}}{w}_i^0{\[{\mathrm{\α}}_k{A}_i/\left({\mathrm{\α}}_w{w}_i^0\right)\]}^{1/({\mathrm{\α}}_k+{\mathrm{\α}}_w)}}---\left(11\right)$

Formula (12) substitutes into formula (10) and obtains：

$t_i=\frac{(\stackrel{\‾}{W}-\underset{i\∉I_a}{\mathrm{\Σ}}{t}_i^{{\mathrm{\α}}_w}{w}_i^0){\[{\mathrm{\α}}_k{A}_i/\left({\mathrm{\α}}_w{w}_i^0\right)\]}^{1/({\mathrm{\α}}_k+{\mathrm{\α}}_w)}}{\underset{i\∈I_a}{\mathrm{\Σ}}{w}_i^0{\[{\mathrm{\α}}_k{A}_i/\left({\mathrm{\α}}_w{w}_i^0\right)\]}^{1/({\mathrm{\α}}_k+{\mathrm{\α}}_w)}}---\left(12\right)$

In view of the Operations of Interva Constraint of topological variable, that is,：

$t_i^{(k+1)}=\left\{\begin{array}{cc}\underset{\&OverBar;}{t_i}& (t_i<\underset{\&OverBar;}{t_i})\\ t_i& (\underset{\&OverBar;}{t_i}<t_i<1)\\ 1& (t_i>1)\end{array}\right.---\left(13\right).$

3. the tree topology based on OPTIMIZATION OF CONTINUUM STRUCTURES as claimed in claim 2 creates structure method, and its feature exists In, update and actively collect, return again to calculate t by formula (12)_i, so circulate, until actively collecting constant terminates circulation, try to achieve optimal solution t^*Also the optimal solution of formula (5) is, structure is changed by formula (1), into circulating next time, such iteration is until satisfaction convergence is accurate Then：

Δ e=| (e^(k+1)-e^(k))/e^(k+1)|≤ε(14)

Wherein, e^(k)And e^(k+1)It is front-wheel and the structure total strain energy of epicycle iteration, ε is convergence precision, and ε=0.001 is taken herein.

4. the tree topology based on OPTIMIZATION OF CONTINUUM STRUCTURES is created described in a kind of utilization claim 1~3 any one Tree-shaped supporting structure in the building hall that structure method is proposed.

5. the tree topology based on OPTIMIZATION OF CONTINUUM STRUCTURES is created described in a kind of utilization claim 1~3 any one Tree-shaped supporting structure in the airport building that structure method is proposed.

6. the tree topology based on OPTIMIZATION OF CONTINUUM STRUCTURES is created described in a kind of utilization claim 1~3 any one Tree-shaped supporting structure in the high-speed railway waiting hall that structure method is proposed.

7. the tree topology based on OPTIMIZATION OF CONTINUUM STRUCTURES is created described in a kind of utilization claim 1~3 any one Tree-shaped supporting structure in the large public activity center that structure method is proposed.