CN113722859B - Uncertainty structure static response determination method based on convex polyhedron model - Google Patents
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Abstract
The invention discloses an uncertainty structure static response determination method based on a convex polyhedron model. The method describes the distribution domain of the uncertain parameters with a convex polyhedron. Compared with the ultra-long square model and the ultra-ellipsoid model, the convex polyhedron model has smaller area and more accurate corresponding response space. The first order perturbation analysis is carried out at the center of the convex polyhedron, the linear expansion is carried out on the convex polyhedron, and the solution theorem of the static structural response domain under the condition of uncertainty of the convex polyhedron is obtained. Finally, an example shows that compared with an ultra-long square model and an ultra-ellipsoid model, the method can obtain a smaller response area and keep higher precision.
Description
Technical Field
The invention relates to the technical field of structural uncertainty propagation analysis, in particular to an uncertainty structural static response determination method based on a convex polyhedron model. The method is based on a convex polyhedron model, takes uncertainty such as load, material, structural dimension parameters and the like into consideration, and achieves uncertainty propagation analysis on displacement response of a structure through first-order perturbation expansion.
Background
The static response analysis of the structure is the basis for structure optimization and structure performance verification. Thus, accurate structural static response analysis is critical. In the conventional case, the accuracy of the static response analysis is generally ensured by the accuracy of finite element simulation. However, for engineering problems, the structure may contain many uncertain parameters due to material aging, structural damage, manufacturing errors, load environment variations, etc. These uncertainty parameters can lead to non-negligible dispersion of the structural response. In the simplest case, the influence of the uncertainty parameter can be taken into account by a safety factor. But it is clear that the safety factor method is an empirical method that is rough in terms of uncertainty. Thus, if there is an appropriate amount of data, the data-based evaluation method is a more accurate method of evaluating the influence of uncertain parameters.
When sample data is sufficient, probabilistic methods are often the more appropriate method of interpreting uncertainty. The large number of samples ensures the accuracy of the probabilistic approach. However, for practical engineering problems, a large number of samples are often not available due to time and cost constraints. Since the distribution form of the uncertain parameters is generally unknown, when the number of samples is insufficient, it is difficult not only to determine the distribution form of the samples but also to accurately estimate the values of the distributed parameters. To solve this problem, a non-probabilistic method that does not depend on a large amount of data has been proposed. Unlike probabilistic methods, non-probabilistic methods use simple geometries, such as a hyper-ellipsoid and a super-long cube, to quantify the distribution of samples.
In the last few decades, a number of non-probabilistic methods have been proposed, ben Haim first introducing the structural non-probabilistic security concept based on convex models. After that, researchers have further developed non-probabilistic methods and have been widely applied to the problems of structural reliability optimization, reliability analysis, uncertainty propagation analysis, and the like. Convex models have two main forms: one is a hyper-ellipsoidal model, and the other is a hyper-cuboid model or a section set model. In the two-dimensional case, the hyper-ellipsoidal model is degenerated to an ellipse, while the hyper-ellipsoidal model is degenerated to a rectangle, both of which are simple forms for describing uncertainty.
In constructing the hyper-ellipsoids and hyper-cubes based on sample data, all sample data is typically included. Notably, non-probabilistic methods based on hyper-cubes and hyper-ellipsoids inevitably introduce problems of quantified dilation due to the simplicity of the geometric form. When describing the sample distribution using a hyper-ellipsoidal model or a hyper-cuboid model, some additional area is typically covered. In order to avoid the problems, the invention adopts a convex polyhedron model to quantize the sample. Compared with the ultra-ellipsoidal model and the ultra-cuboid model, the convex polyhedron model adopts the minimum convex hull to represent uncertainty, so that a tighter sample convex set and fewer redundant areas can be realized. Although a convex polyhedron can tightly surround a sample, it is difficult to solve the static response of a structure in the case of a convex polyhedron model due to its complex and irregular geometry. In order to solve the problems, the invention provides a vertex solving mode of uncertain response under a convex polyhedron model based on first order perturbation expansion.
Disclosure of Invention
The invention aims to solve the technical problems that: the method can calculate the static response of the uncertainty structure more accurately compared with the static response determination method of the uncertainty structure based on the ultra-ellipsoidal model or the ultra-cuboid model.
The invention adopts the technical scheme that: the uncertainty structure static response determination method based on the convex polyhedron model can carry out propagation analysis on the structure static response with the convex polyhedron uncertainty parameter, consider the uncertainty of load, material modulus, processing technology and the like in actual engineering, and describe the uncertainty of a structure rigidity matrix and a load vector by adopting the convex polyhedron model. Compared with the traditional ultra-ellipsoidal model and ultra-cuboid model, the convex polyhedron model has smaller area, and can avoid the problem of expansion in the uncertainty propagation analysis process. And then, performing first order perturbation expansion based on the central value of the convex polyhedron, and performing linearization simplification on the propagation problem, and finally establishing the vertex solving theorem of the uncertain response under the convex polyhedron model.
The implementation steps of the invention are as follows:
step one: establishing a static equilibrium equation in a finite element form aiming at a structure containing uncertainty parameters of a convex polyhedron;
step two: determining the distribution range of the total stiffness matrix and the total load vector in the static equilibrium equation in the step one by utilizing the convex polyhedron;
step three: performing first order perturbation expansion on the overall rigidity matrix and the load vector at the center of the convex polyhedron to obtain the overall rigidity matrix and the overall load vector after the first order perturbation expansion;
step four: and establishing a balance equation by using the overall stiffness matrix and the overall load vector after the first order perturbation expansion, and calculating to obtain the distribution range of the displacement response, thereby obtaining the uncertainty structural static response of the convex polyhedron model.
In the first step, the static equilibrium equation is established as follows:
where K is the overall stiffness matrix, u is the overall displacement vector, f is the overall load vector,for a convex set of the overall stiffness matrix K distribution, +.>Is a convex set of the overall load vector f distribution.
In the second step, the equation of the convex polyhedron is:
wherein K is k K=1, 2, …, M and f l L=1, 2, …, N are the vertex global stiffness matrix and the vertex global load vector, respectively, M is the number of vertex global stiffness matrices, N is the number of vertex global load vectors, α k And beta l Is a coefficient, and M and N are the number of vertex global stiffness matrices and vertex global load vectors, respectively.
In the third step, the following form of first order perturbation expansion is adopted:
K k ＝K 0 +ΔK k (10)
f l ＝f 0 +Δf l (11)
wherein K is 0 For a nominal overall stiffness matrix, ΔK k K=1, 2, …, M is the perturbation overall stiffness matrix, M is the number of perturbation overall stiffness matrices, f 0 As a nominal overall load vector Δf l L=1, 2, …, N is the perturbation total load vector, N is the number of perturbation total load vectors.
In the fourth step, the established equilibrium equation is:
K 0 for the nominal overall stiffness matrix, M is the number of perturbing overall stiffness matrices, N is the number of perturbing overall load vectors, Δf l L=1, 2, …, N is the perturbation overall load vector, α k And beta l Is a coefficient with a value range of [0,1 ]]，u 0 For a nominal overall displacement vector, deltau is a perturbation overall displacement vector.
Compared with the prior art, the invention has the advantages that: compared with an uncertainty structure static response determining method based on an ultra-ellipsoidal model or an ultra-cuboid model, the method can obtain a smaller uncertainty area, can slow down the expansion problem in the uncertainty propagation analysis process, and avoids the overlarge obtained structure static response range; the method is simple and easy to implement, and provides a new theoretical method for static response analysis of an uncertainty structure.
Drawings
FIG. 1 is a schematic flow diagram of a method for determining an uncertainty structural static response based on a convex polyhedron model of the present invention;
FIG. 2 is a schematic diagram of load and boundary conditions for an embodiment of the present invention, wherein (a) is a truss structure diagram and (b) is a cross-sectional view of truss members;
FIG. 3 is a comparison of the calculation results of different methods according to the embodiments of the present invention;
FIG. 4 is sample data of elastic modulus for an embodiment of the present invention;
FIG. 5 is sample data of load and cross-section parameters for an embodiment of the present invention;
FIG. 6 is a graph showing the numerical comparison of the calculation results of the different methods according to the embodiments of the present invention.
Detailed Description
The invention is further described below with reference to the drawings and specific examples.
As shown in fig. 1, the uncertainty structure static response determination method based on the convex polyhedron model can obtain smaller response area and keep higher precision compared with an ultra-long polyhedron model and an ultra-ellipsoidal model.
The method specifically comprises the following steps:
step one: for a structure containing uncertainty parameters, establishing a static equilibrium equation in a finite element form, and adopting a convex set to determine the distribution range of an overall stiffness matrix and an overall load vector:
where K is the overall stiffness matrix, u is the overall displacement vector, f is the overall load vector,is a convex set distributed by a rigidity matrix K>A convex set distributed for the load vector f;
step two: describing a convex set in the form of a convex polyhedronAnd->
Wherein K is i I=1, 2, …, M and f i I=1, 2, …, N is the vertex global stiffness matrix and vertex global load vector, α k And beta l Is a coefficient, and M and N are the number of vertex global stiffness matrices and vertex global load vectors, respectively.
Step three: performing first order perturbation expansion on the vertex global stiffness matrix and the vertex global load vector at a nominal value:
K k ＝K 0 +ΔK k (k＝1,2,…,M) (16)
f l ＝f 0 +Δf l (l＝1,2,…,N) (17)
wherein K is 0 For a nominal overall stiffness matrix, ΔK k To perturb the overall stiffness matrix, f 0 As a nominal overall load vector Δf l To perturb the overall load vector.
Step four: and writing the balance direction into the following form by using the unfolded vertex total stiffness matrix and the vertex total load vector:
wherein u is 0 For the nominal displacement vector, deltau is the perturbation displacement vector.
Step five: using equationsAnd->Formula (18) can be reduced to:
due to K 0 u 0 ＝f 0 Thus, formula (19) can be reduced to:
step six: ignoring the higher-order term ΔK in equation (20) k Δu, obtainable:
further organized by formula (21) is:
step seven: multiplying on both sides of (22)Let->γ kl ＝α k β l Finally, the method comprises the following steps:
wherein gamma is kl As a coefficient, deltau k,l Is the nominal stiffness with a load (Δf) l -ΔK k u 0 ) The displacement response at that time.
Step eight: finally, the relation u=u is used 0 And (5) obtaining the convex polyhedron distributed by the displacement vector u after the convex polyhedron of the delta u is obtained through calculation.
Examples:
as shown in fig. 2 (a), consider a planar truss structure, each transverse of the trussThe length of the rod and the vertical rod is 1m. The rod is L-shaped in cross section with a thickness h=0.01m, as shown in fig. 2 (b). The joints at the upper and lower ends of the left side of the truss are fixed, and a vertical downward load F is applied to the middle joint at the right end of the truss y And a load F horizontally rightward x . The displacement response to be analyzed is the horizontal displacement u at the load node x And vertical displacement u y 。
Assuming modulus of elasticity E, horizontal load F x Vertical load F y Section parameter L x And L y Are uncertainty parameters whose sample values are listed in fig. 4 and 5, and these sample data are used for subsequent uncertainty structural static response calculations.
According to the sample data, the minimum ellipsoid model of the load sample can be calculated as follows:
the minimum hyper-cuboid model of the load sample is:
{(F x ,F y )|F x ∈[4.4940×10 4 ,5.4524×10 4 ],F y ∈[-1.1137×10 5 ,-9.3559×10 4 ]} (25)
in addition, the minimum ellipsoid model of the cross-section parameters is:
the minimum hyper-cuboid model of the section parameters is:
{(L x ,L y )|L x ∈[0.0904,0.1116],L y ∈[0.0885,0.1110]} (27)
and the minimum ellipsoid model and the minimum hyper-cuboid model of the elastic modulus are degraded into interval forms, which are:
E∈[2.0511×10 11 ,2.1362×10 11 ]Pa(28)
respectively utilizing the method provided by the inventionAnd calculating the horizontal displacement u by Monte Carlo simulation, an ellipsoid model and an ultra-cuboid model x And vertical displacement u y The results may be plotted as shown in fig. 3 and the data for the results as shown in fig. 6. From fig. 3 and 6, the following can be seen:
(1) The convex polyhedron (degenerated into a convex polygon in two dimensions) calculated by the method of the present invention contains a large portion of Monte Carlo simulation samples. Specifically, the convex polygon contained 97.40% of the simulated samples. Since the simulated samples are based on an accurate distribution function, it can be considered that the convex polygon contains 97.40% of the true sample points, already with sufficient accuracy for engineering problems.
(2) As shown in fig. 6, the accuracy of the ultra-long square model and the ultra-ellipsoidal model is slightly higher than that of the convex polyhedron model (97.40%) →98.99%,97.40% → 98.36%). However, the calculated envelope area of the ultra-long square model and the ultra-ellipsoidal model is much larger than that of the convex polyhedron model (100% →141.87%,100% → 120.73%). Therefore, the convex polyhedron model can effectively avoid overestimation of the response compared to the conventional ultra-long square model and the ultra-ellipsoidal model, and at the same time, it loses accuracy only slightly with respect to the ultra-cuboid model and the ultra-ellipsoidal model.
(3) Since the convex polyhedron model uses the smallest convex set of samples to quantify uncertainty, it contains less area than and is a subset of the ultra-long cube model and the ultra-ellipsoid model. Therefore, the response range calculated by the convex polyhedron model must be within the response range calculated by the ultra-long square model and the ultra-ellipsoidal model, as shown in fig. 3.
The above is only a specific step of the present invention, and does not limit the protection scope of the present invention; the method can be widely applied to the field of static structure uncertainty analysis based on convex polyhedron models, and all the technical schemes formed by adopting equivalent transformation or equivalent replacement fall within the scope of the invention.
The present invention is not described in detail in part as being well known to those skilled in the art.
Claims (4)
1. The uncertainty structure static response determining method based on the convex polyhedron model is characterized by comprising the following implementation steps:
step one: establishing a static equilibrium equation in a finite element form aiming at a structure containing uncertainty parameters of a convex polyhedron;
step two: determining the distribution range of the total stiffness matrix and the total load vector in the static equilibrium equation in the step one by utilizing the convex polyhedron;
step three: performing first order perturbation expansion on the overall rigidity matrix and the load vector at the center of the convex polyhedron to obtain the overall rigidity matrix and the overall load vector after the first order perturbation expansion;
step four: establishing a balance equation by using the overall stiffness matrix and the overall load vector after the first order perturbation expansion, and calculating to obtain the distribution range of the displacement response, thereby obtaining the uncertainty structure static response of the convex polyhedron model;
in the first step, the static equilibrium equation is established as follows:
where K is the overall stiffness matrix, u is the overall displacement vector, f is the overall load vector,for a convex set of the overall stiffness matrix K distribution, +.>Is a convex set of the overall load vector f distribution.
2. The method for determining the uncertainty structural static response based on the convex polyhedron model according to claim 1, wherein: in the second step, the equation of the convex polyhedron is:
wherein K is k K=1, 2, …, M and f l L=1, 2, …, N are the vertex global stiffness matrix and the vertex global load vector, respectively, M is the number of vertex global stiffness matrices, N is the number of vertex global load vectors, α k And beta l Is a coefficient, and M and N are the number of vertex global stiffness matrices and vertex global load vectors, respectively.
3. The method for determining the uncertainty structural static response based on the convex polyhedron model according to claim 1, wherein: in the third step, the following form of first order perturbation expansion is adopted:
K k ＝K 0 +ΔK k (4)
f l ＝f 0 +Δf l (5)
wherein K is 0 For a nominal overall stiffness matrix, ΔK k K=1, 2, …, M is the perturbation overall stiffness matrix, M is the number of perturbation overall stiffness matrices, f 0 As a nominal overall load vector Δf l L=1, 2, …, N is the perturbation total load vector, N is the number of perturbation total load vectors.
4. The method for determining the uncertainty structural static response based on the convex polyhedron model according to claim 1, wherein: in the fourth step, the established equilibrium equation is:
K 0 for the nominal overall stiffness matrix, M is the number of perturbed overall stiffness matrices, N is the number of perturbed overall load vectorsQuantity Δf l L=1, 2, …, N is the perturbation overall load vector, α k And beta l Is a coefficient with a value range of [0,1 ]]，u 0 For a nominal overall displacement vector, deltau is a perturbation overall displacement vector.
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| CN108710737A (en) * | 2018-05-10 | 2018-10-26 | 北京航空航天大学 | One kind containing crowned design argument structure static displacement boundary method based on linear programming assessment |
| CN110803296A (en) * | 2019-10-15 | 2020-02-18 | 中国直升机设计研究所 | Helicopter full-mechanical static test lift force system dummy |
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| CN108121865A (en) * | 2017-12-15 | 2018-06-05 | 北京航空航天大学 | Analysis method is propagated in a kind of response of the structure section containing more uncertain parameters based on adjoint variable method |
| CN108710737A (en) * | 2018-05-10 | 2018-10-26 | 北京航空航天大学 | One kind containing crowned design argument structure static displacement boundary method based on linear programming assessment |
| CN110803296A (en) * | 2019-10-15 | 2020-02-18 | 中国直升机设计研究所 | Helicopter full-mechanical static test lift force system dummy |
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