CN101510231A - Plate case structural bead distribution design method based on root forming mechanism - Google Patents

Abstract

The invention relates to a plate-shell structure stiffener distribution design method based on plant root formation mechanism. On the basis of analyzing and summarizing the plant root formation mechanism, a distribution design method of plate-shell structure stiffener is summed up and divided into three processes: initialization, growth and differentiation. If excessive unreasonable stiffeners are differentiated in the growth process, degradation happens in the follow-up process. Therefore, the design method leads the structure to intelligently develop towards the structure with optimal mechanical properties when the stiffeners grows.

Description

Form the plate case structural bead distribution design method of mechanism based on root system of plant

Technical field

The present invention relates to a kind of construction design method, particularly a kind of plate case structural bead distribution design method that forms mechanism based on root system of plant.

Background technology

Although coming in every shape of occurring in nature root system of plant, main Gene Handling by plant itself, but the root system of kindred plant has different forms in different environment, illustrate that plant root species form is determined by the growth environment of himself to a great extent, and the formation of its form has the function of adaptive Growth environment.These adaptive Growth rules can reduce bends gravity, hydrotaxis and haptotropism etc.The form formation of root system of plant under the growth rule control of these adaptive Growth environment, by means such as growth, difference and degenerations, is rationally adjusted the growth direction and the growth rate of branch just dexterously, reaches the optimizing process of allomeric function optimum.

If the distribution of plate and shell structure reinforcement is regarded as a process that progressively forms, the deviser wishes that this forming process can be adaptive to certain condition of work so, makes structure progressively tend to have the structure of optimum mechanical property under the constraint of material volume.The such process and the forming process of plant root species form have the similarity of essence: certain condition of work is the growing environment in the plant root species form forming process, and reinforcement distributes and then distributes for Root morphology.If by means of the distribution that the formation mechanism of plant root species form is come layout board shell structure reinforcement, design the structure that obtains so and have and the similar optimality of root system of plant: energy self-adaptation condition of work, and reach optimum mechanical property.

Summary of the invention

The distribution and the root system of plant that The present invention be directed to the plate and shell structure reinforcement form the problem that mechanism has the similarity of essence, a kind of plate case structural bead distribution design method that forms mechanism based on root system of plant has been proposed, on the research basis of analyzing and summing up root system of plant formation mechanism, sum up plate case structural bead distribution design method, method for designing makes structure in the growth reinforcement, is tending towards having the structure of optimum mechanical property intelligently.

Technical scheme of the present invention is: a kind of plate case structural bead distribution design method based on root system of plant formation mechanism comprises the steps:

(1) at first based structures being dispersed is 8 node quadrilateral shell units, and reinforcement is made up of the adjacent node of substrate casing unit, is the beam element of 2 nodes, and the initial cross-section of each reinforcement is long-pending to be A ₀

(2) according to the support and the carrying condition of based structures, select the node conduct " seed " on several based structures, after selected " seed ", " seed " is comprised in the set of branch point { among the P}, and the reinforcement that " seed " can grow up all around be comprised in growth reinforcement set among the R}, the difference critical value A of given volume rate of growth Δ V, reinforcement simultaneously _bAnd the cumulative volume upper limit V of reinforcement _Limit

(3) developmental process: total is carried out finite element analysis, and to be included in growth reinforcement set the reinforcement of growing up among the P} is according to the design sensitivity of following formula calculating target function with respect to sectional area,

The maximum stiffness design:

$\frac{\∂U}{\∂A_j}=\frac{1}{2}{F}^T\frac{\∂u}{\∂A_j};$ $\frac{\∂u}{\∂A_j}=-K^{-1}\frac{\∂K}{\∂A_j}u,$ In the formula, U-structure total strain energy, the nodal displacement vector of u-structure, the panel load vector of F-structure, the integral rigidity matrix of K-structure, A _jBe the sectional area of reinforcement j,

Maximum natural frequency of vibration design:

$\frac{\∂(1/{\mathrm{\λ}}_1)}{\∂A_j}=u_1^T(\frac{1}{{\mathrm{\λ}}_1}\frac{\∂M}{\∂A_j}-\frac{1}{{\mathrm{\λ}}_1^2}\frac{\∂K}{\∂A_j})u_1,$ In the formula, λ ₁The base of-structure the circular frequency of shaking; The global stiffness matrix of K-structure, M-architecture quality matrix, u ₁The modal vector that-base shakes,

Under the control of volume growth rate, set the reinforcement among the R} promptly upgrades its sectional area by following formula according to himself design sensitivity growth or degeneration,

The maximum stiffness design:

$\mathrm{\Δ}{A}_j=\frac{\mathrm{\ΔV}(\frac{\∂U}{\∂A_j}-w\frac{\stackrel{\‾}{\∂U}}{\∂A})}{\underset{i=1}{\overset{m}{\mathrm{\Σ}}}\frac{\∂U}{\∂A_i}{l}_i-w\frac{\stackrel{\‾}{\∂U}}{\∂A}\underset{i=1}{\overset{m}{\mathrm{\Σ}}}{l}_i},$ In the formula, Be the average design sensitivity of structural strain energy, l _iBe the length of reinforcement i, w is the weighting coefficient of CONTROL VOLUME increment, and Δ V is the volume growth rate, and m is the quantity of the reinforcement of growing up,

Maximum natural frequency of vibration design:

$\mathrm{\Δ}{A}_j=\frac{\mathrm{\ΔV}(\frac{\∂(1/{\mathrm{\λ}}_1)}{\∂A_j}-w\frac{\stackrel{\‾}{\∂(1/{\mathrm{\λ}}_1)}}{\∂A})}{\underset{i=1}{\overset{m}{\mathrm{\Σ}}}\frac{\∂(1/{\mathrm{\λ}}_1)}{\∂A_i}{l}_i-w\frac{\stackrel{\‾}{\∂(1/{\mathrm{\λ}}_1)}}{\∂A}\underset{i=1}{\overset{m}{\mathrm{\Σ}}}{l}_i},$ In the formula,

Be the shake average design sensitivity of circular frequency of structure base;

(4) difference process: if the sectional area after a reinforcement upgrades is more than or equal to difference critical value A _bThe time, two end points of this reinforcement are considered to have the ability of difference, and join the set of branch point { among the P} as new branch point, and and all reinforcements of joining of new branch point join the set of growth reinforcement { among the R}, enter step (3) and carry out growing up in next step of growing up or degenerating, and reduce to the long-pending A of initial cross-section of reinforcement when the sectional area of a reinforcement ₀The time, this reinforcement is considered to degenerate fully, { remove the P}, and its two end points also { is removed the P} from set from set;

(5) carry out repeatedly step (3) and (4), and until the reinforcement cumulative volume of having grown up, promptly the summation of reinforcement area and length product reaches predetermined cumulative volume upper limit V _Limit

Described " seed " should be chosen in and support on border and the loading position, satisfies the principle of the transfer route basically identical of the distribution of reinforcement and load.

Beneficial effect of the present invention is: the present invention is based on the plate case structural bead distribution design method that root system of plant forms mechanism, come the distribution of layout board shell structure reinforcement by means of the formation mechanism of plant root species form, make structure in the growth reinforcement, be tending towards having the structure of optimum mechanical property intelligently.

Description of drawings

Fig. 1 the present invention is based on the plate case structural bead distribution design method design flow diagram that root system of plant forms mechanism;

The reinforcement distribution plan that Fig. 2 has maximum stiffness during for the square plate bearing plane uniformly distributed load of the fixing band center pit of the present invention's two opposite side;

The reinforcement distribution plan that Fig. 3 has maximum stiffness when bearing the center line uniformly distributed load for the square plate of the fixing band center pit of the present invention's two opposite side;

The reinforcement distribution plan that Fig. 4 has maximum stiffness when bearing torsional load for the square plate of the fixing band center pit of the present invention's two opposite side;

The reinforcement distribution plan that Fig. 5 has maximum stiffness when bearing the bending combined load for the square plate of the fixing band center pit of the present invention's two opposite side;

Fig. 6 is the distributed mode graphoid of vein of the present invention;

Fig. 7 designs a model for the fixing square plate of the present invention's two adjacent sides;

The reinforcement distribution plan that Fig. 8 has the maximum natural frequency of vibration for the fixing square plate of the present invention's two adjacent sides;

Fig. 9 square panel vibration mode figure that maximum natural frequency of vibration reinforcement distributes for the present invention's two adjacent sides fixedly have.

Embodiment

1, the description of method design:

Research thin-walled part shell structure, symmetric arrangement the negligible original gusset of sectional dimension on the top and bottom of this plate and shell structure.In order to obtain having the structure of optimal mechanical properties, the optimal design problem can be described as: under the constraint condition of the given reinforcement cumulative volume upper limit, ask the area of each reinforcement, make the mechanical property optimum of structure.That is:

Maximum/littleization: obj

Be tied in V≤V _Limit

Design variable: A _j(j=1,2,3 ..., n)

Wherein, obj is a design object, can select according to designing requirement in plate and shell structure, as the maximum stiffness or the maximum natural frequency of vibration of structure; V is the reinforcement cumulative volume of finally having grown up; V _LimitIt is the given reinforcement cumulative volume upper limit; A _jSectional area for reinforcement j; N is the quantity of reinforcement.

2, the realization of method design:

Design cycle is divided into following 3 steps as shown in Figure 1:

The A initialization procedure:

(1) at first based structures being dispersed is 8 node quadrilateral shell units, and reinforcement is made up of the adjacent node of substrate casing unit, is the beam element of 2 nodes.The initial cross-section of each reinforcement is long-pending to be very little, the negligible A0 of numerical value.

(2), select the node conduct " seed " on several based structures according to the support and the carrying condition of based structures.After selected " seed ", " seed " is comprised in the set of branch point { among the R}, and the reinforcement that " seed " can grow up all around is comprised in the set of growth reinforcement { among the P}.The difference critical value A of while given volume rate of growth Δ V, reinforcement _bAnd the cumulative volume upper limit V of reinforcement _Limit

The B developmental process:

Total is carried out finite element analysis, and { reinforcement of growing up among the R} is calculated as follows the design sensitivity of objective function obj with respect to sectional area to being included in the set of growth reinforcement.

The maximum stiffness design:

$\frac{\∂U}{\∂A_j}=\frac{1}{2}{F}^T\frac{\∂u}{\∂A_j}---\left(1\right)$

$\frac{\∂u}{\∂A_j}=-K^{-1}\frac{\∂K}{\∂A_j}u---\left(2\right)$

In the formula, U---structure total strain energy, the nodal displacement vector of u---structure, the panel load vector of F---structure, the integral rigidity matrix of K---structure.

Maximum natural frequency of vibration design:

$\frac{\∂(1/{\mathrm{\λ}}_1)}{\∂A_j}=u_1^T(\frac{1}{{\mathrm{\λ}}_1}\frac{\∂M}{\∂A_j}-\frac{1}{{\mathrm{\λ}}_1^2}\frac{\∂K}{\∂A_j})u_1---\left(3\right)$

In the formula, λ ₁---the base of the structure circular frequency of shaking; The global stiffness matrix of K---structure, M---architecture quality matrix, u ₁---the modal vector that base shakes.

Under the control of volume growth rate, { reinforcement among the R} promptly upgrades its sectional area, wherein l by (4) and (5) formula according to himself design sensitivity growth or degeneration in set _iBe the length of reinforcement i, w is the weighting coefficient of CONTROL VOLUME increment, and Δ V is the volume growth rate, and m is the quantity of the reinforcement of growing up.The following formula explanation, the rate of growth of each reinforcement, promptly the increment of sectional area is proportional with respect to the design sensitivity of sectional area with the objective function obj of structure.When the design sensitivity of a certain reinforcement during greater than mean value, the sectional area of this reinforcement increases, otherwise, then reduce.Hence one can see that, and design sensitivity has determined the growth or the degeneration of each reinforcement.

The maximum stiffness design:

$\mathrm{\Δ}{A}_j=\frac{\mathrm{\ΔV}(\frac{\∂U}{\∂A_j}-w\frac{\stackrel{\‾}{\∂U}}{\∂A})}{\underset{i=1}{\overset{m}{\mathrm{\Σ}}}\frac{\∂U}{\∂A_i}{l}_i-w\frac{\stackrel{\‾}{\∂U}}{\∂A}\underset{i=1}{\overset{m}{\mathrm{\Σ}}}{l}_i}---\left(4\right)$

In the formula,

--the average design sensitivity of-structural strain energy.

Maximum natural frequency of vibration design:

$\mathrm{\Δ}{A}_j=\frac{\mathrm{\ΔV}(\frac{\∂(1/{\mathrm{\λ}}_1)}{\∂A_j}-w\frac{\stackrel{\‾}{\∂(1/{\mathrm{\λ}}_1)}}{\∂A})}{\underset{i=1}{\overset{m}{\mathrm{\Σ}}}\frac{\∂(1/{\mathrm{\λ}}_1)}{\∂A_i}{l}_i-w\frac{\stackrel{\‾}{\∂(1/{\mathrm{\λ}}_1)}}{\∂A}\underset{i=1}{\overset{m}{\mathrm{\Σ}}}{l}_i}---\left(5\right)$

In the formula,

--the shake average design sensitivity of circular frequency of-structure base.

C difference process:

If the sectional area after a certain reinforcement upgrades is more than or equal to difference critical value A _bThe time, two end points of this reinforcement are considered to have the ability of difference, and as new branch point join branch point set among the P}, and and all reinforcements of joining of new branch point join the set of growth reinforcement { among the R}, so that in next grows up the step, grow up or degenerate.And the sectional area of working as a certain reinforcement reduces to the long-pending A of the initial cross-section of reinforcement ₀The time, this reinforcement is considered to degenerate fully, { remove the R}, and its two end points also { is removed the P} from set from set.

Step B (growth) and C (difference) carry out repeatedly, reaches the predetermined cumulative volume upper limit until the reinforcement cumulative volume of having grown up.

3, the method design parameter is analyzed:

In the implementation procedure of algorithm, the selection of design parameter has certain influence to design result, and design parameter mainly comprises " seed " and difference critical value.The present invention has carried out detail analysis and has proposed reasonable suggestions the selection of design parameter.

1) selection of " seed ":

" seed " of root system of plant has only one usually, and main root is directly from " seed ", according to adaptive Growth rule difference square root and hair root; And the plate and shell structure in the engineering is because the complicacy of boundary condition, and how often not only one in " seed " of reinforcement selects the number of " seed " and distribution " seed " directly to influence design result.Studies show that, because in fact the distribution of reinforcement answers basically identical with the transfer route of load, therefore " seed " should be chosen in and support on border and the loading position, satisfies such distribution principle, structure just can be tending towards optimum under the control of adaptive Growth rule.

2) difference critical value A _bSelection:

If the sectional area after a certain reinforcement upgrades is more than or equal to difference critical value A _bThe time, two end points of this reinforcement are considered to have the ability of difference, and obviously the size of difference critical value may exert an influence to design result.Studies show that less difference critical value can produce more branch, it is comparatively complicated that structural bead is distributed; Otherwise, then may produce less branch.The deviser can select according to layout strategy.But, because method for designing based on the root system of plant growth mechanism, the reinforcement of not only can growing up, and the reinforcement of having grown up is degenerated, if therefore difference too much irrational reinforcement, can in follow-up developmental process, degenerate, therefore as long as the difference critical value is chosen in certain scope, the distribution of main reinforcement is not exerted an influence basically.

4, method design application examples

The method for designing that the present invention proposes can be carried out the design of the maximum stiffness and the maximum natural frequency of vibration to the plate and shell structure of arbitrary shape and boundary condition, below with several typical examples as an illustration.

Fig. 2,3,4,5 is that the square plate of the fixing band center pit of two opposite side is when bearing different transverse loads, with the reinforcement distribution design result of maximum stiffness as design object.The thickness of plate and length of side ratio are made as 0.001.Wherein Fig. 2 and 3 bears bending load, but is respectively face uniformly distributed load and center line uniformly distributed load.And Fig. 4 is for bearing torsional load, and Fig. 5 is for bearing the bending combined load.According to different load, selected different " seed ", shown in stain among each figure, notice that " seed " all is selected in loading place and support place.The cumulative volume upper limit of reinforcement is set at V _Limit=0.25V0.As seen from the figure, 1/4 on four jiaos of (Fig. 2) reinforcement slave plate under the effect of face uniformly distributed load and support edge is set out, grows up in the middle of plate.And under other three kinds of situations, reinforcement is only four of slave plate jiaos of growth of setting out all.Under the effect of bending combined load (Fig. 5), the volume of reinforcement deflection is distributed in the bigger side of load.Their final strain energy is respectively U ₁=4.18 * 10 ^-3U ₀, U ₁=3.25 * 10 ^-3U ₀, U ₁=3.08 * 10 ^-3U ₀, and U ₁=2.06 * 10 ^-3U ₀

Fig. 6 is a foliate lamina, is used for simulating the distribution of vein.According to the weight distribution of leaf and actual fixation case, suppose plate above bear uniform transverse load, and fix at the following center section of plate.The cumulative volume upper limit of reinforcement is set at V _Limit=0.25V ₀, " seed " is selected in the following bearing of plate.The distribution of the reinforcement that obtains as seen from Figure 6, extremely is similar to actual vein and distributes.Therefore the distribution of vein has the function that increases the leaf integral rigidity really.

Fig. 7,8,9 be two adjacent sides fix other both sides freely the ratio of thickness and the length of side be 0.001 square plate, it is carried out with the maximum natural frequency of vibration is the reinforcement distribution design of objective function.Fig. 7 is for designing a model, and Fig. 8 is a design result.Admittedly aid the border areas with " C " expression among the figure, with " F " expression free margins.9 some conducts " seed " that are distributed on the fixed edge have been selected.As seen from the figure, main reinforcement is from the 1/4 of fixed edge, and along growing up with the direction of fixed edge quadrature, the direction that is folded to free angle is then extended.When the reinforcement volume is increased to 0.6V ₀The time, the base circular frequency of shaking finally is increased to 4218.9 λ ₀, simultaneously, mode of oscillation figure as shown in Figure 9, the mode of oscillation of plate also changes along with the growth of reinforcement, and peak swing reduces gradually.

Claims (2)

1, a kind of plate case structural bead distribution design method based on root system of plant formation mechanism comprises the steps:

(1) at first based structures being dispersed is 8 node quadrilateral shell units, and reinforcement is made up of the adjacent node of substrate casing unit, is the beam element of 2 nodes, and the initial cross-section of each reinforcement is long-pending to be A ₀

(2) according to the support and the carrying condition of based structures, select the node conduct " seed " on several based structures, after selected " seed ", " seed " is comprised in the set of branch point { among the P}, and the reinforcement that " seed " can grow up all around be comprised in growth reinforcement set among the R}, the difference critical value A of given volume rate of growth Δ V, reinforcement simultaneously _bAnd the cumulative volume upper limit V of reinforcement _Limit

(3) developmental process: total is carried out finite element analysis, and to be included in growth reinforcement set the reinforcement of growing up among the R} is according to the design sensitivity of following formula calculating target function with respect to sectional area,

The maximum stiffness design:

$\frac{\∂U}{\∂A_j}=\frac{1}{2}{F}^T\frac{\∂u}{\∂A_j};$ $\frac{\∂u}{\∂A_j}=-K^{-1}\frac{\∂K}{\∂A_j}u,$ In the formula, U-structure total strain energy, the nodal displacement vector of u-structure, the panel load vector of F-structure, the integral rigidity matrix of K-structure, A _jBe the sectional area of reinforcement j,

Maximum natural frequency of vibration design:

$\frac{\∂(1/{\mathrm{\λ}}_1)}{\∂A_j}=u_1^T(\frac{1}{{\mathrm{\λ}}_1}\frac{\∂M}{\∂A_j}-\frac{1}{{\mathrm{\λ}}_1^2}\frac{\∂K}{\∂A_j})u_1,$ In the formula, λ ₁The base of-structure the circular frequency of shaking; The global stiffness matrix of K-structure, M-architecture quality matrix, u ₁The modal vector that-base shakes,

Under the control of volume growth rate, set the reinforcement among the R} promptly upgrades its sectional area by following formula according to himself design sensitivity growth or degeneration,

The maximum stiffness design:

$\mathrm{\Δ}{A}_j=\frac{\mathrm{\ΔV}(\frac{\∂U}{\∂A_j}-w\frac{\stackrel{\‾}{\∂U}}{\∂A})}{\underset{i=1}{\overset{m}{\mathrm{\Σ}}}\frac{\∂U}{\∂A_i}{l}_i-w\frac{\stackrel{\‾}{\∂U}}{\∂A}\underset{i=1}{\overset{m}{\mathrm{\Σ}}}{l}_i},$ In the formula,

Be the average design sensitivity of structural strain energy, l _iBe the length of reinforcement i, w is the weighting coefficient of CONTROL VOLUME increment, and Δ V is the volume growth rate, and m is the quantity of the reinforcement of growing up,

Maximum natural frequency of vibration design:

$\mathrm{\Δ}{A}_j=\frac{\mathrm{\ΔV}(\frac{\∂(1/{\mathrm{\λ}}_1)}{\∂A_j}-w\frac{\stackrel{\‾}{\∂(1/{\mathrm{\λ}}_1)}}{\∂A})}{\underset{i=1}{\overset{m}{\mathrm{\Σ}}}\frac{\∂(1/{\mathrm{\λ}}_1)}{\∂A_i}{l}_i-w\frac{\stackrel{\‾}{\∂(1/{\mathrm{\λ}}_1)}}{\∂A}\underset{i=1}{\overset{m}{\mathrm{\Σ}}}{l}_i},$ In the formula,

Be the shake average design sensitivity of circular frequency of structure base;

(4) difference process: if the sectional area after a reinforcement upgrades is during more than or equal to difference critical value Ab, two end points of this reinforcement are considered to have the ability of difference, and join the set of branch point { among the P} as new branch point, and and all reinforcements of joining of new branch point join the set of growth reinforcement { among the R}, enter step (3) and carry out growing up in next step of growing up or degenerating, and reduce to the long-pending A of initial cross-section of reinforcement when the sectional area of a reinforcement ₀The time, this reinforcement is considered to degenerate fully, { remove the R}, and its two end points also { is removed the P} from set from set;

(5) carry out repeatedly step (3) and (4), and until the reinforcement cumulative volume of having grown up, promptly the summation of reinforcement area and length product reaches predetermined cumulative volume upper limit V _Limit

2, according to the described plate case structural bead distribution design method that forms mechanism based on root system of plant of claim 1, it is characterized in that, described " seed " should be chosen in and support on border and the loading position, satisfies the principle of the transfer route basically identical of the distribution of reinforcement and load.

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