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CN118940341B - Transverse shear stiffness estimation method, homogenization simulation method, system and medium - Google Patents

Transverse shear stiffness estimation method, homogenization simulation method, system and medium Download PDF

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CN118940341B
CN118940341B CN202411419143.6A CN202411419143A CN118940341B CN 118940341 B CN118940341 B CN 118940341B CN 202411419143 A CN202411419143 A CN 202411419143A CN 118940341 B CN118940341 B CN 118940341B
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CN118940341A (en
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印长俊
谢思雨
李卓
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Xiangtan University
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Abstract

The invention provides a transverse shear stiffness estimation method, a homogenization simulation method, a system and a medium, wherein the method is suitable for a periodical shell, and by introducing a coarse scale coordinate system and a fine scale coordinate system, the method for calculating the transverse shear stiffness of the periodic shell is obtained by utilizing the progressive homogenization theory of the shell, and the coefficient of the transverse shear stiffness matrix E of the periodic shell meeting the displacement form is determined by a classical homogenization process. The method is suitable for periodic shells containing any inclusions or boundaries, such as hollow floors, sandwich structures, waffle structures, plate-grid structures and the like, and is also suitable for transverse shear stiffness calculation of homogeneous thick shells and laminated shells. Based on the method, the obtained transverse shear rigidity is further substituted into a finite element model of the homogenizing shell, the homogenizing simulation can be carried out on the periodical shell, accurate displacement and internal force are obtained, the accurate displacement and internal force are obtained with extremely low calculation cost and calculation precision loss, and the engineering design requirement is met.

Description

Transverse shear stiffness estimation method, homogenization simulation method, system and medium
Technical Field
The invention relates to the technical field of performance detection of composite shells, in particular to a transverse shear stiffness estimation method, a homogenization simulation system and a medium, which are suitable for periodic shells.
Background
The composite shell structure is a novel material formed by combining multiple layers of materials in a laminating mode, has remarkable advantages in the aspects of light weight, high strength, rigidity, temperature resistance, corrosion resistance, flexible design and the like, and has wide application prospects in the fields of aerospace engineering, automobile industry, ship and ocean engineering, chemical engineering, wind power generation, modern building engineering, construction and the like.
The performance of the composite shell structure is influenced by the level of the composition layers, including composition materials, material interfaces, geometric shapes, axial force effects, adjacent supports and the like, and the performance of the shell structure such as stress, strain, failure, damage and the like of the layer can be measured and estimated through experimental, numerical simulation or theoretical deduction methods. In engineering practice, the transverse shear rigidity of the shell has remarkable influence on the mechanical property and behavior of the shell, and the ratio of the shear stress to the displacement caused by the relative displacement of the shell structure under the action of external force along the normal direction is shown, so that the capability of resisting deformation of the shell structure under the action of transverse force is shown, and the method has important significance for designing the shell structures such as lightweight structural members, building prefabricated members and the like by knowing and determining the transverse shear rigidity of the shell no matter the design of the uniform thick shell, the laminated shell and the periodic shell structure is realized.
Currently, there are theories and methods related to calculating transverse shear stiffness for homogenous thick shells and laminated shells. Representative voxel methods (REPRESENTATIVE VOLUME ELEMENT, RVE) and asymptotic homogenization methods (asymptotic homogenization, AH) are two numerical methods for predicting periodic material equivalent properties. The representative voxel method is a numerical homogenization algorithm constructed based on an average field theory, and the equivalent mechanical parameters of the representative voxels are calculated by using a mesomechanics analysis means through selecting the representative voxels and used for macroscopic solution of the problem, and the average mechanical properties of the representative voxels are used for representing the performance of the whole structure. The representative voxel method has clear mechanical concept, is simple to operate, is suitable for homogenizing equivalent estimation similar to a honeycomb material structure, and is not suitable for numerical calculation of a complex periodic shell structure. The asymptotic homogenization method has a strict mathematical basis, and mature theory and algorithm exist in the aspect of calculating the material equivalent performance of the three-dimensional and two-dimensional periodic microstructure, but for periodic shell structures such as hollow floors, sandwich structures, waffle structures, plate-lattice structures and the like, no theory and method for calculating transverse shear stiffness exist, and although A.A. Kalamkarov et al propose the asymptotic homogenization mathematical theory of the periodic plate-shell structure, the asymptotic homogenization mathematical theory still cannot realize finite element realization and macroscopic homogenization simulation of the periodic shell structure.
Disclosure of Invention
In view of the drawbacks and deficiencies of the prior art, according to a first aspect of the object of the present invention, a method is provided for estimating the transverse shear stiffness of a periodical shell, which is adapted to the periodical shell by using the progressive homogenization theory of the shell, and is adapted to the estimation of the transverse shear stiffness of a periodical shell containing any inclusions or boundaries, such as hollow floors, sandwich structures, waffle structures, plate-lattice structures, etc., and is also adapted to the estimation of the transverse shear stiffness of homogeneous thick shells, laminated shells.
A method of estimating transverse shear stiffness according to the object of the first aspect described above, suitable for use with a periodic shell, comprising the steps of:
Step 1, establishing a coarse-scale coordinate system (alpha 1,α2, gamma) and a fine-scale coordinate system (y 1,y2, z) for representing a periodic shell mathematical model, wherein the coarse-scale coordinate system (alpha 1,α2, gamma) is an orthogonal curve coordinate system, and a relation is established between the coarse-scale coordinate system (alpha 1,α2, gamma) and the fine-scale coordinate system (y 1,y2, z) based on the thickness of a shell under the coarse-scale coordinate system (alpha 1,α2, gamma) and the single-cell transverse dimension of the shell under the fine-scale coordinate system (y 1,y2, z);
Step 2, constructing a relative displacement function of the shell by utilizing a perturbation method under a coarse scale coordinate system (alpha 1,α2 and gamma) and a fine scale coordinate system (y 1,y2 and z);
Step 3, according to the strain displacement relation of the shell under the orthogonal curve coordinate system, obtaining a strain function based on the relative displacement of the shell based on the relative displacement of alpha 1,α2 and gamma directions, and determining a strain tensor corresponding to the perturbation coefficient;
step 4, obtaining a stress function based on relative displacement of the shell according to the strain displacement relation of the shell under an orthogonal curve coordinate system;
Step 5, inputting the strain tensor corresponding to the perturbation coefficient in the step 3 into a generalized Hooke's law of three-dimensional elastic mechanics to obtain the stress tensor corresponding to the perturbation coefficient;
step 6, inputting a balance equation of the shell to obtain a balance equation corresponding to the perturbation expansion coefficient according to the stress function based on the relative displacement of the shell in the step 4;
step 7, inputting a balance equation corresponding to the perturbation expansion coefficient in the step 6 based on the stress tensor corresponding to the perturbation expansion coefficient in the step 5, and obtaining a balance equation which is expressed by relative displacement and takes the transverse shear deformation into consideration;
Step 8, inputting the displacement form of the shell into the balance equation taking the transverse shear deformation into consideration in the step 7, and obtaining a control equation in a domain omega;
Step 9, determining a control equation in a domain theta based on the control equation in the domain omega;
And step 10, converting a control equation in the domain theta into a finite element form, carrying out numerical value solving, and obtaining coefficients of the transverse shear stiffness matrix E of the periodic shell meeting the displacement form by using a classical homogenization process.
According to a second aspect of the object of the present invention, a homogenizing simulation method taking into account the transverse shear stiffness is provided, applicable to periodic shells, having little calculation cost and calculation accuracy loss, applicable to periodic shells having any hollow inclusions or complex boundaries.
A homogenization simulation method taking into account transverse shear stiffness according to the above object of the second aspect, applicable to a periodic shell, comprising the steps of:
Establishing a finite element model of a unit cell of a periodic shell, inputting material parameters of the unit cell, applying a periodic boundary to the unit cell, and applying an initial stress load to the unit cell;
Solving the finite element result of the unit cell according to the method to obtain the coefficient of the transverse shear stiffness matrix E of the periodic shell;
and (3) establishing a homogenized finite element model of the periodic shell, inputting a transverse shear stiffness matrix E, and calculating the central deflection of the periodic shell after applying boundary conditions and loads.
According to a third aspect of the object of the present invention there is provided a computer system comprising one or more processors and a memory storing instructions operable, when executed by the one or more processors, to cause the one or more processors to perform operations comprising performing the processes of the aforementioned methods.
According to a fourth aspect of the object of the present invention there is provided a computer readable medium storing a computer program comprising instructions/instruction sets executable by one or more processors, which instructions/instruction sets, when executed by the one or more processors, implement the processes of the aforementioned methods.
According to a fifth aspect of the object of the present invention, there is provided a computer program product comprising a computer program which, when executed by a processor, implements the steps of the aforementioned method.
It should be understood that all combinations of the foregoing concepts, as well as additional concepts described in more detail below, may be considered a part of the inventive subject matter of the present disclosure as long as such concepts are not mutually inconsistent. In addition, all combinations of claimed subject matter are considered part of the disclosed inventive subject matter.
The foregoing and other aspects, embodiments, and features of the present teachings will be more fully understood from the following description, taken together with the accompanying drawings. Other additional aspects of the invention, such as features and/or advantages of the exemplary embodiments, will be apparent from the description which follows, or may be learned by practice of the embodiments according to the teachings of the invention.
Drawings
The drawings are not intended to be drawn to scale. In the drawings, each identical or nearly identical component that is illustrated in various figures may be represented by a like numeral. For purposes of clarity, not every component may be labeled in every drawing.
FIG. 1 is a flow chart of a method of estimating lateral shear stiffness of a periodic shell in accordance with an embodiment of the present invention.
FIG. 2 is a flow chart of a periodic shell homogenization simulation in accordance with an embodiment of the present invention.
FIG. 3 is a schematic illustration of an isopipe having a periodic distribution of square inclusions in a slab according to one embodiment of the present invention.
Fig. 4 is a schematic view of a homogenization shell according to an embodiment of the invention.
Fig. 5 is a schematic diagram of simulation calculation results through DNS and HNS under an ANSYS platform according to an embodiment of the present invention.
Fig. 6 is a schematic diagram of simulation calculation results through DNS and HNS under an ABAQUS platform according to an embodiment of the present invention.
Detailed Description
For a better understanding of the technical content of the present invention, specific examples are set forth below, along with the accompanying drawings.
Aspects of the invention are described in this disclosure with reference to the drawings, in which are shown a number of illustrative embodiments. The embodiments of the present disclosure are not necessarily intended to include all aspects of the invention. It should be understood that the various concepts and embodiments described above, as well as those described in more detail below, may be implemented in any of a number of ways, as the disclosed concepts and embodiments are not limited to any implementation. Additionally, some aspects of the disclosure may be used alone or in any suitable combination with other aspects of the disclosure.
The method is suitable for periodic shells, the theory and the method for calculating the transverse shear rigidity of the periodic shells are obtained by utilizing the progressive homogenization theory of the shells, the method is suitable for periodic shells containing any inclusions or boundaries, such as hollow floors, sandwich structures, waffle structures, plate-grid structures and the like, and meanwhile, the method is also suitable for estimating the transverse shear rigidity of uniform thick shells and laminated shells and obtaining the transverse shear rigidity. Based on the method, the obtained transverse shear rigidity is further substituted into a finite element model of the homogenizing shell, so that homogenizing simulation considering the transverse shear rigidity can be realized on the periodical shell, accurate displacement and internal force can be obtained with extremely low calculation cost and calculation precision loss, and the engineering design requirement is met.
{ Example 1}
The transverse shear stiffness estimation method according to the disclosed embodiments of the invention is applicable to periodic shells, i.e., the transverse shear stiffness estimation method of periodic shells, can be executed by a computer system. The computer system can be realized by a terminal, a server and a cloud server, and can also be realized by an interactive system of the terminal and the server.
In this embodiment, taking a computer system with a terminal as an example, the method for estimating the transverse shear stiffness of the periodic shell in combination with the example shown in fig. 1 includes the following steps:
Step 1, establishing a coarse-scale coordinate system (alpha 1,α2, gamma) and a fine-scale coordinate system (y 1,y2, z) for representing a periodic shell mathematical model, wherein the coarse-scale coordinate system (alpha 1,α2, gamma) is an orthogonal curve coordinate system, and the relation between the coarse-scale coordinate system (alpha 1,α2, gamma) and the fine-scale coordinate system (y 1,y2, z) is established based on the thickness of a shell under the coarse-scale coordinate system (alpha 1,α2, gamma) and the single-cell transverse dimension of the shell under the fine-scale coordinate system (y 1,y2, z);
Step 2, constructing a relative displacement function of the shell by utilizing a perturbation method under a coarse scale coordinate system (alpha 1,α2 and gamma) and a fine scale coordinate system (y 1,y2 and z);
Step 3, according to the strain displacement relation of the shell under the orthogonal curve coordinate system, obtaining a strain function based on the relative displacement of the shell based on the relative displacement of alpha 1,α2 and gamma directions, and determining a strain tensor corresponding to the perturbation coefficient;
step 4, obtaining a stress function based on relative displacement of the shell according to the strain displacement relation of the shell under an orthogonal curve coordinate system;
Step 5, inputting the strain tensor corresponding to the perturbation coefficient in the step 3 into a generalized Hooke's law of three-dimensional elastic mechanics to obtain the stress tensor corresponding to the perturbation coefficient;
step 6, inputting a balance equation of the shell to obtain a balance equation corresponding to the perturbation expansion coefficient according to the stress function based on the relative displacement of the shell in the step 4;
step 7, inputting a balance equation corresponding to the perturbation expansion coefficient in the step 6 based on the stress tensor corresponding to the perturbation expansion coefficient in the step 5, and obtaining a balance equation which is expressed by relative displacement and takes the transverse shear deformation into consideration;
Step 8, inputting the displacement form of the shell into the balance equation taking the transverse shear deformation into consideration in the step 7, and obtaining a control equation in a domain omega;
Step 9, determining a control equation in a domain theta based on the control equation in the domain omega;
And 10, converting a control equation in the domain theta into a finite element form, solving, and obtaining coefficients of a transverse shear stiffness matrix E of the periodic shell meeting the displacement form by using a classical homogenization process.
In an embodiment of the present invention, a coarse-scale coordinate system (α 1,α2, γ) and a fine-scale coordinate system (y 1,y2, z) are introduced, where the coarse-scale coordinate system is an orthogonal curve coordinate system, let α= [ α 1,α2],y= [y1,y2 ].
In the embodiment of the invention, for convenience of explanation, we define the rule that the subscript of the variable is expressed by Greek letters to be valued in the {1, 2} range, and the subscript of the variable is expressed by Latin letters to be valued in the {1, 2, 3} range.
As an alternative embodiment, in step 1, a relation is established based on the thickness of the lower shell in the coarse-scale coordinate system (α 1,α2, γ) and the unit cell lateral dimensions of the lower shell in the fine-scale coordinate system (y 1,y2, z), specifically as follows:
;
Wherein h 1,h2 represents the lateral dimensions of the shell unit cell in the directions alpha 1 and alpha 2, respectively, delta represents the perturbation coefficient, and delta represents the thickness of the shell in a coarse-scale coordinate system.
It will be appreciated that in embodiments of the invention, the definition of unit cell is the same as that in the composite and reinforced elements of construction document of a.a. Kalamkarov et al, referring to a microstructure that is periodically repeated in a periodic shell structure.
As an alternative embodiment, in step 2, the displacement function of the shell is obtained using perturbation methods in a coarse-scale coordinate system (α 1,α2, γ) and in a fine-scale coordinate system (y 1,y2, z), expressed as follows:
;
where u 1,u2 and u 3 represent relative displacements in the alpha 1,α2 and gamma directions, respectively. Refers to the infinitely small higher order of the device,And is a variable.
As an alternative embodiment, in step 3, according to the strain displacement relation of the shell under the orthogonal curve coordinate system, a strain function based on the relative displacement of the shell is obtained based on the relative displacement in the α 1,α2 and γ directions, and a strain tensor corresponding to the perturbation coefficient is determined, including:
substituting the relative displacement function of the shell into the strain displacement relation of the shell under an orthogonal curve coordinate system to obtain the strain function based on the relative displacement of the shell, wherein the strain function is expressed as follows:
;
Wherein, Representing a strain tensor; represents the strain tensor corresponding to the perturbation expansion coefficient δ l, l e {0, 1,2,. };
and obtaining a strain tensor corresponding to delta 2 according to the strain function based on the relative displacement of the shell The expression is as follows:
;
Wherein u 1,u2,u3 represents relative displacements in the alpha 1,α2 and gamma directions, respectively, Representing the displacement corresponding to perturbation expansion coefficient delta l, i e {1, 2, 3}, l e {0, 1,2, 3. };、 The coefficients of the first quadratic form of the respective orthogonal curve coordinate systems (α 1,α2, γ).
As an alternative embodiment, in step 4, a stress function based on the relative displacement of the shell is obtained according to the strain displacement relationship of the shell under the orthogonal curvilinear coordinate system, expressed as follows:
;
Wherein, Representing a stress tensor; Represents the stress tensor corresponding to the perturbation expansion coefficient δ l, l e {0, 1, 2. Refers to the infinitely small higher order of the device,And is a variable.
As an alternative embodiment, in step 5, the foregoing obtaining a stress tensor corresponding to the perturbation coefficient is expressed as:
;
In the formula, Represents the stress tensor corresponding to the perturbation expansion coefficient δ l, l e {0, 1,2,. }; Representing a fourth-order material stiffness tensor under a fine-scale coordinate system, wherein y= [ y 1,y2 ]; representation and representation AndAn associated strain tensor; representation and representation Related strain tensors.
Further, a stress tensor corresponding to delta 2 can be obtainedThe expression is as follows:
;
;
In the formula, 、The principal curvature functions of the curved surfaces are respectively represented, the independent variable is alpha 1,α2, for a constant curvature curved surface,、Is constant.
As an alternative embodiment, in step 6, the stress function based on the relative displacement of the shell in step 4 is input into the balance equation of the shell to obtain the balance equation corresponding to the perturbation expansion coefficient, namely the balance equation of δ -1,δ0 and δ 1 is obtained, specifically as follows:
A first procedure: ;
the second equation: ;
third procedure: ;
In the formula, AndAll represent differential operators.
It should be understood that the equilibrium equation for the shell herein refers in particular to the equilibrium equation for the shell as defined in the document composite and reinforced elements of construction by a.a. Kalamkarov et al.
Further, in step 7, the stress tensor corresponding to the perturbation coefficient in step 5 is input into the balance equation corresponding to the perturbation expansion coefficient in step 6, and a balance equation taking into account the transverse shear deformation expressed in terms of relative displacement is obtained. As an alternative embodiment, specifically, the stress tensor corresponding to δ 2 is input into the foregoing third procedure, and a balance equation expressed by relative displacement and considering transverse shear deformation is obtained, where the equation is expressed as:
;
In the formula, ,,Respectively representing differential operators.
As an alternative embodiment, in step 8, the displacement form of the shell is expressed as:
;
In the formula, 、Representing the displacement field hypothesis functions, respectively.
Further, the displacement form of the shell is input to the balance equation taking into account the transverse shear deformation in the foregoing step 7, and a control equation in a domain Ω is obtained, which is expressed as follows:
;
In the formula, Normal vector representing upper and lower boundaries of unit cell in domain ΩThe j-direction component of (c).The differential operator is represented by a differential operator,A hypothetical function representing the stress tensor,AndIs a component expression thereof; Omega refers to the domain under the coordinate system (ζ 1,ξ2, z), ∈{1,2},i∈{1, 2, 3},j∈{1, 2, 3}。
As an alternative embodiment, in step 9, based on the control equation in the domain Ω, a control equation in a domain Θ is determined, which is expressed as follows:
;
In the formula, ,Θ refers to the domain in the coordinate system (η 1,η2,η3), η= [ η 1,η2,η3];d= [d1,d2, 0],d1 and d 2 represent the lengths of the domain Θ in the directions η 1 and η 2, respectively; Representing the vertex of the domain theta.
As an alternative embodiment, in step 10, the control equation in the field Θ is converted into a finite element form expressed as follows:
;
In the formula, ,Respectively refers to the strain matrix of the discrete units in the domain Θ; An overall stiffness matrix representing domain Θ; representing the equivalent overall load vector of domain Θ; In the representation domain Θ Is a numerical solution to (a);
the coefficients of the transverse shear stiffness matrix E of the periodic shell satisfying the displacement form are obtained by using a classical homogenization process, expressed as follows:
;
representing a classical homogenization process.
{ Example 2}
As shown in fig. 2, according to the present disclosure, a homogenizing simulation method considering transverse shear stiffness is further provided, which is applicable to a periodic shell, that is, a homogenizing simulation method considering transverse shear stiffness, where the periodic shell refers to a periodic shell (that is, a homogenizing shell) having an equal thickness and a principal curvature equal to a constant assumption, and the homogenizing simulation method of the periodic shell includes the following steps:
Establishing a finite element model of a unit cell of a periodic shell, inputting material parameters of the unit cell, applying a periodic boundary to the unit cell, and applying an initial stress load to the unit cell;
Solving the finite element result of the unit cell according to the numerical solution method to obtain the coefficient of the transverse shear stiffness matrix E of the periodic shell;
and (3) establishing a homogenized finite element model of the periodic shell, inputting a transverse shear stiffness matrix E, and calculating the central deflection of the periodic shell after applying boundary conditions and loads.
{ Example 3}
Based on the embodiments 1 and 2, the following description will further take an example of an equal-thickness shell with square inclusions distributed periodically in the plate as shown in fig. 3 to 6.
Referring to fig. 3, where (a), (b), (c), and (d) represent a plan view and a sectional view of an equal-thickness shell, and a plan view and a sectional view of a unit cell, respectively. As shown in FIGS. 3 (a) and (b), the periodic shell was an equal-thickness shell having a square inclusion in a plate with periodic distribution, and had a length and width of 9.4m and a thickness of 0.3m.
The engineering constants of the matrix material of the plate are as follows, young's modulus E m =25.5 GPa, poisson's ratio v m =0.2.
In connection with fig. 3 (a), there are square inclusions distributed periodically in the plate.
In combination with (c) and (d) of fig. 3, each square inclusion was 0.814m long and 0.2m thick.
The engineering constants of the inclusion materials are as follows, young's modulus Ei∈{0, 0.2, 0.4, 0.6, 0.8, 1.0, 1.2, 1.4, 1.6, 1.8, 2.0, 2.2, 2.4, 2.6, 2.9, 3.0, 3.2, 3.4, 3.6, 3.8, 4.0}Em, poisson ratio v m =0.2.
As shown in FIGS. 3 (c) and (d), the volume of inclusions in the unit cell of the shell was 50%.
An equispaced load of 10kPa was applied to the top surface of the plate.
In this example, we used conventional Direct Numerical Simulation (DNS) and 3 Homogenized Numerical Simulations (HNSs) based on the method of the present invention to perform simulation calculations, the parameters and environments used for the simulation calculations are shown in table 1 below. The homogenization shell used for the equalization numerical simulation is shown in fig. 4, where (a) and (b) of fig. 4 represent a plan and cross-sectional representation of the homogenization shell, respectively.
Referring to fig. 5, a comparison of simulation calculation results of DNS and HNS methods under the ANSYS platform is shown. Wherein the center deflection comparison of the shell is shown in (a) of fig. 5, the bending moment comparison of the sections A and B of the shell under the condition of four sides simple support is shown in (B) of fig. 5, the bending moment comparison of the sections A and B of the shell under the condition of four sides solid support is shown in (c) of fig. 5, and the bending moment comparison of the sections A and B of the shell under the condition of four corners point support is shown in (d) of fig. 5. As can be seen from the comparison of fig. 5, the method of the invention substitutes the transverse shear stiffness matrix E as a necessary condition into the central deflection of the equivalent homogenizing shell obtained by simulation calculation, and compared with the central deflection of the direct numerical simulation of the periodical shell, the central deflection of the homogenizing shell maintains high consistency, and the error is not more than 5%.
Referring to fig. 6, a comparison of simulation calculation results of DNS and HNS methods under ABAQUS platform is shown. The center deflection contrast of the shell under the four-side simple support condition is shown in (a) of fig. 6, wherein a green line is an analytic solution of a classical plate-shell theory (CPT) and a first-order shear deformation theory (FSDT), the center deflection contrast of the shell under the four-side solid support condition is shown in (b) of fig. 6, and the center deflection contrast of the shell under the four-corner point support condition is shown in (c) of fig. 6. As can be seen from the comparison of fig. 6, the method of the invention substitutes the transverse shear stiffness matrix E as a necessary condition into the central deflection of the equivalent homogenizing shell obtained by simulation calculation, and compared with the central deflection simulated by the traditional classical plate shell theory (CPT) and the first-order shear deformation theory (FSDT) of the periodic shell, the method of the invention keeps high consistency and extremely low error.
Meanwhile, by combining the comparison of the simulation environment and the calculation time, the method can greatly reduce the requirements on calculation memory and calculation time, calculate and obtain accurate uniform shell displacement and internal force with extremely low calculation cost and calculation precision loss, and meet the requirements of engineering design.
{ Example 4}
In one exemplary embodiment, a computer system is provided, which may be a server.
As one example, a computer device includes a processor, memory, input/Output interfaces (I/O), and a communication interface. The processor, the memory and the input/output interface are connected through a system bus, and the communication interface is connected to the system bus through the input/output interface. The processor acts as a processing core for data, applications, and provides computing and control capabilities. The memory, and in particular the shell, employs a combination of non-volatile memory and cache. The nonvolatile memory is used to store, among other things, operating system programs, applications, and databases that are executed by the processor. It should be appreciated that caches are typically configured to be integrated with the processor.
It should be appreciated that the above example structures are merely exemplary of partial structures typical of a computer system and are not limiting of the computer system to which the aspects of the present disclosure may be applied, and that in one or more embodiments, a computer system may include more or fewer components, or may combine certain components, or have different arrangements of components.
In one embodiment, the present disclosure also provides a computer system comprising one or more processors and memory. The memory can be used to store instructions that are operable, when executed by one or more processors, to cause the one or more processors to perform operations comprising performing the processes of the periodic shell transverse shear stiffness estimation method or the homogenization simulation method of the foregoing embodiments.
{ Example 5}
In one embodiment, the present disclosure also provides a computer readable medium storing software comprising instructions executable by one or more computers that when executed by the one or more computers perform the processes of the periodic shell transverse shear stiffness estimation method or homogenization simulation method of the foregoing embodiments.
{ Example 6}
In one embodiment, the present disclosure also provides a computer program product comprising a computer program which, when executed by a processor, implements the process of the periodic shell transverse shear stiffness estimation method or homogenization simulation method of the foregoing embodiments.
While the invention has been described with reference to preferred embodiments, it is not intended to be limiting. Those skilled in the art will appreciate that various modifications and adaptations can be made without departing from the spirit and scope of the present invention. Accordingly, the scope of the invention is defined by the appended claims.

Claims (12)

1.一种横向剪切刚度估计方法，适用于周期性壳，其特征在于，所述方法包括以下步骤：1. A method for estimating transverse shear stiffness, applicable to periodic shells, characterized in that the method comprises the following steps: 步骤1、建立用于表征周期性壳数学模型的粗尺度坐标系(α 1, α 2, γ)和细尺度坐标系(y 1, y 2, z)，所述粗尺度坐标系(α 1, α 2, γ)为正交曲线坐标系，所述粗尺度坐标系(α 1,α 2, γ) 与细尺度坐标系(y 1, y 2, z)之间基于粗尺度坐标系(α 1, α 2, γ)下壳的厚度以及细尺度坐标系(y 1, y 2, z)下壳的单胞横向尺寸建立关系；Step 1, establishing a coarse-scale coordinate system ( α 1 , α 2 , γ ) and a fine-scale coordinate system ( y 1 , y 2 , z ) for characterizing a periodic shell mathematical model, wherein the coarse-scale coordinate system ( α 1 , α 2 , γ ) is an orthogonal curvilinear coordinate system, and a relationship is established between the coarse-scale coordinate system ( α 1 , α 2 , γ ) and the fine-scale coordinate system ( y 1 , y 2 , z ) based on the thickness of the shell under the coarse-scale coordinate system ( α 1 , α 2 , γ ) and the lateral size of the unit cell of the shell under the fine-scale coordinate system ( y 1 , y 2 , z ); 步骤2、在粗尺度坐标系(α 1, α 2, γ) 与细尺度坐标系(y 1, y 2, z)下，利用摄动法构建出壳的相对位移函数；Step 2: In the coarse-scale coordinate system ( α 1 , α 2 , γ ) and the fine-scale coordinate system ( y 1 , y 2 , z ), the relative displacement function of the shell is constructed using the perturbation method; 步骤3、根据在正交曲线坐标系下壳的应变位移关系，基于α 1, α 2, γ方向的相对位移获得基于壳的相对位移的应变函数，并确定与摄动系数相应的应变张量；Step 3, according to the strain-displacement relationship of the shell in the orthogonal curvilinear coordinate system, based on the relative displacements in the α 1 , α 2 , γ directions, a strain function based on the relative displacement of the shell is obtained, and the strain tensor corresponding to the perturbation coefficient is determined; 步骤4、根据在正交曲线坐标系下壳的应变位移关系，获得基于壳的相对位移的应力函数；Step 4: According to the strain-displacement relationship of the shell in the orthogonal curvilinear coordinate system, a stress function based on the relative displacement of the shell is obtained; 步骤5、将所述步骤3中与摄动系数相应的应变张量输入三维弹性力学的广义胡克定律，得到与摄动系数相应的应力张量；Step 5, inputting the strain tensor corresponding to the perturbation coefficient in step 3 into the generalized Hooke's law of three-dimensional elasticity to obtain the stress tensor corresponding to the perturbation coefficient; 步骤6、根据所述步骤4中基于壳的相对位移的应力函数，输入壳的平衡方程得到与摄动展开系数相应的平衡方程；Step 6: According to the stress function based on the relative displacement of the shell in step 4, the equilibrium equation of the shell is input to obtain the equilibrium equation corresponding to the perturbation expansion coefficient; 步骤7、基于所述步骤5中与摄动系数相应的应力张量，输入所述步骤6中与摄动展开系数相应的平衡方程，获得以相对位移表示的考虑横向剪切变形的平衡方程；Step 7, based on the stress tensor corresponding to the perturbation coefficient in step 5, the equilibrium equation corresponding to the perturbation expansion coefficient in step 6 is input to obtain the equilibrium equation considering the transverse shear deformation expressed in relative displacement; 步骤8、将壳的位移形式输入到所述步骤7中的考虑横向剪切变形的平衡方程，获得在域Ω中的控制方程；Step 8, input the displacement form of the shell into the equilibrium equation considering the transverse shear deformation in step 7 to obtain the control equation in the domain Ω; 步骤9、基于所述域Ω中的控制方程，确定域Θ中的控制方程；Step 9: Based on the control equation in the domain Ω, determine the control equation in the domain Θ; 步骤10、将所述域Θ中的控制方程转换为有限元形式，并进行求解，利用经典均匀化过程获得满足所述位移形式的周期性壳的横向剪切刚度矩阵E的系数。Step 10: Convert the control equations in the domain Θ into finite element form and solve them, and use the classical homogenization process to obtain the coefficients of the transverse shear stiffness matrix E of the periodic shell that satisfies the displacement form. 2.根据权利要求1所述的横向剪切刚度估计方法，其特征在于，所述步骤1中，所述基于粗尺度坐标系(α 1, α 2, γ)下壳的厚度以及细尺度坐标系(y 1, y 2, z)下壳的单胞横向尺寸建立关系，具体如下：2. The method for estimating lateral shear stiffness according to claim 1, characterized in that, in the step 1, the relationship between the thickness of the lower shell in the coarse-scale coordinate system ( α 1 , α 2 , γ ) and the lateral size of the unit cell of the lower shell in the fine-scale coordinate system ( y 1 , y 2 , z ) is established as follows: ； ; 其中，h 1, h 2分别表示壳的单胞在α 1和α 2方向的横向尺寸；δ表示摄动系数，代表粗尺度坐标系中壳的厚度。Among them, h 1 and h 2 represent the lateral dimensions of the unit cell of the shell in the α 1 and α 2 directions respectively; δ represents the perturbation coefficient, which represents the thickness of the shell in the coarse-scale coordinate system. 3.据权利要求1所述的横向剪切刚度估计方法，其特征在于，所述步骤3中，所述根据在正交曲线坐标系下壳的应变位移关系，基于α 1, α 2, γ方向的相对位移获得基于壳的相对位移的应变函数，并确定与摄动系数相应的应变张量，包括：3. The method for estimating transverse shear stiffness according to claim 1, characterized in that, in step 3, the step of obtaining a strain function based on the relative displacement of the shell based on the relative displacement in the α 1 , α 2 , and γ directions according to the strain-displacement relationship of the shell in the orthogonal curvilinear coordinate system, and determining a strain tensor corresponding to the perturbation coefficient comprises: 将壳的相对位移函数代入在正交曲线坐标系下壳的应变位移关系，获得基于壳的相对位移的应变函数，表达如下：Substituting the relative displacement function of the shell into the strain-displacement relationship of the shell in the orthogonal curvilinear coordinate system, the strain function based on the relative displacement of the shell is obtained, which is expressed as follows: ； ; 其中，表示应变张量；表示与摄动展开系数δ l 相对应的应变张量，l∈{0, 1,2, …}；in, represents the strain tensor; represents the strain tensor corresponding to the perturbation expansion coefficient δ l , l ∈ {0, 1,2, …}; 并且，根据所述基于壳的相对位移的应变函数，获得δ 2对应的应变张量，表达如下：And, according to the strain function based on the relative displacement of the shell, the strain tensor corresponding to δ 2 is obtained: , expressed as follows: ； ; 其中，u 1, u 2, u 3分别表示α 1, α 2和γ方向的相对位移，表示与摄动展开系数δ l 相对应的位移， i∈{1, 2, 3}；l∈{0, 1, 2, 3, …}；、分别正交曲线坐标系(α 1, α 2,γ)第一个二次型的系数。Where u 1 , u 2 , u 3 represent the relative displacements in the α 1 , α 2 and γ directions respectively. represents the displacement corresponding to the perturbation expansion coefficient δ l , i ∈ {1, 2, 3}; l ∈ {0, 1, 2, 3, …}; , The coefficients of the first quadratic form of the orthogonal curvilinear coordinate system ( α 1 , α 2 , γ ) respectively. 4.根据权利要求1所述的横向剪切刚度估计方法，其特征在于，所述步骤5中，所述得到与摄动系数相应的应力张量，表达为：4. The method for estimating transverse shear stiffness according to claim 1, characterized in that, in step 5, the stress tensor corresponding to the perturbation coefficient is obtained and expressed as: ； ; 式中，表示与摄动展开系数δ l 相对应的应力张量，l∈{0, 1, 2, …}；表示细尺度坐标系下的四阶材料刚度张量，y = [y 1, y 2]；表示与和相关的应变张量；表示与相关的应变张量；In the formula, represents the stress tensor corresponding to the perturbation expansion coefficient δ l , l ∈{0, 1, 2, …}; represents the fourth-order material stiffness tensor in the fine-scale coordinate system, y = [ y 1 , y 2 ]; Representation and and The associated strain tensor; Representation and The associated strain tensor; 其中，δ 2对应的应力张量，表达如下：Among them, the stress tensor corresponding to δ 2 is , expressed as follows: ； ; ； ; 式中，、分别表示曲面的主曲率函数，自变量是α 1, α 2。In the formula, , They represent the principal curvature functions of the surface, and the independent variables are α 1 and α 2 . 5.根据权利要求4所述的横向剪切刚度估计方法，其特征在于，所述步骤7中，所获得的以相对位移表示的考虑横向剪切变形的平衡方程表达为：5. The method for estimating transverse shear stiffness according to claim 4, characterized in that, in step 7, the equilibrium equation considering the transverse shear deformation expressed in terms of relative displacement is expressed as: ； ; 式中，,,分别表示微分算子。In the formula, , , represent differential operators respectively. 6.根据权利要求5所述的横向剪切刚度估计方法，其特征在于，所述步骤8中，所述壳的位移形式表示为：6. The method for estimating the transverse shear stiffness according to claim 5, characterized in that in step 8, the displacement form of the shell is expressed as: ; ; 式中，、分别表示位移场假定函数；In the formula, , They represent the assumed functions of displacement field respectively; 将壳的位移形式输入到所述步骤7中的考虑横向剪切变形的平衡方程，获得在域Ω中的控制方程，所述域Ω中的控制方程表达如下：The displacement form of the shell is input into the equilibrium equation considering the transverse shear deformation in step 7 to obtain the control equation in the domain Ω, which is expressed as follows: ; ; 式中，表示域Ω中的单胞的上下边界法向量的j方向的分量；表示微分算子，表示应力张量的假定函数，和是其分量表达式；，Ω指在坐标系(ξ 1, ξ 2, z)下的域，∈{1,2}，i∈{1, 2, 3}，j∈{1, 2, 3}。In the formula, represents the upper and lower boundary normal vectors of the unit cell in the domain Ω The component in the j direction; represents the differential operator, An assumed function representing the stress tensor, and is its component expression; , Ω refers to the domain in the coordinate system ( ξ 1 , ξ 2 , z ), ∈{1,2}, i ∈{1, 2, 3}, j ∈{1, 2, 3}. 7.根据权利要求6所述的横向剪切刚度估计方法，其特征在于，所述步骤9中，基于所述域Ω中的控制方程，确定域Θ中的控制方程，所述域Θ中的控制方程表达如下：7. The method for estimating the transverse shear stiffness according to claim 6, characterized in that, in step 9, based on the control equation in the domain Ω, the control equation in the domain Θ is determined, and the control equation in the domain Θ is expressed as follows: ; ; 式中，，；Θ指在坐标系(η 1, η 2, η 3)下的域；η= [η 1, η 2, η 3]；d= [d 1, d 2, 0]，d 1和d 2分别表示域Θ在η 1和η 2方向的长度；表示域Θ的顶点。In the formula, , ; Θ refers to the domain in the coordinate system ( η 1 , η 2 , η 3 ); η = [ η 1 , η 2 , η 3 ]; d = [ d 1 , d 2 , 0], d 1 and d 2 represent the lengths of the domain Θ in the η 1 and η 2 directions, respectively; denotes the vertices of the domain Θ. 8.根据权利要求7所述的横向剪切刚度估计方法，其特征在于，所述步骤10中，所述域Θ中的控制方程转换为有限元形式表达如下：8. The method for estimating transverse shear stiffness according to claim 7, characterized in that, in step 10, the control equation in the domain θ is converted into a finite element form and expressed as follows: ; ; 式中，,分别是指域Θ中离散单元的应变矩阵；表示域Θ的整体刚度矩阵；表示域Θ的等效整体荷载向量；表示域Θ中的数值解；In the formula, , They refer to the strain matrices of discrete elements in domain Θ respectively; represents the global stiffness matrix of domain Θ; represents the equivalent global load vector of domain Θ; In the domain Θ Numerical solution of ; 利用经典均匀化过程获求解满足所述位移形式的周期性壳的横向剪切刚度矩阵E的系数，表达如下：The coefficients of the transverse shear stiffness matrix E of the periodic shell satisfying the displacement form are obtained by using the classical homogenization process and are expressed as follows: ； ; 其中，表示经典的均匀化过程。in, represents the classical homogenization process. 9.一种考虑横向剪切刚度的均匀化仿真方法，适用于周期性壳，所述周期性壳是指等厚度且主曲率等于常数假定的周期性壳，其特征在于，所述均匀化仿真方法包括以下步骤：9. A homogenization simulation method considering transverse shear stiffness, applicable to periodic shells, wherein the periodic shells are periodic shells with equal thickness and principal curvatures equal to constants, characterized in that the homogenization simulation method comprises the following steps: 建立周期性壳的单胞的有限元模型并输入单胞的材料参数，对单胞施加周期性边界，对单胞施加初始应力荷载；Establish a finite element model of a unit cell of a periodic shell and input the material parameters of the unit cell, apply a periodic boundary to the unit cell, and apply an initial stress load to the unit cell; 根据所述权利要求1-8中任意一项所述的方法求解单胞的有限元结果，获得周期性壳的横向剪切刚度矩阵E的系数；Solve the finite element results of the unit cell according to the method described in any one of claims 1 to 8 to obtain the coefficients of the transverse shear stiffness matrix E of the periodic shell; 建立周期性壳的均匀化有限元模型，输入横向剪切刚度矩阵E，在施加边界条件和载荷后，计算周期性壳的中心挠度。A homogenized finite element model of the periodic shell is established, and the transverse shear stiffness matrix E is input. After applying boundary conditions and loads, the central deflection of the periodic shell is calculated. 10.一种计算机系统，其特征在于，包括：10. A computer system, comprising: 一个或多个处理器；one or more processors; 存储器，存储可被操作的指令，所述指令在通过所述一个或多个处理器执行时使得所述一个或多个处理器执行操作，所述操作包括执行所述权利要求1-9中任意一项所述方法的过程。A memory storing operable instructions, wherein when the instructions are executed by the one or more processors, the one or more processors are caused to perform operations, wherein the operations include the process of executing the method described in any one of claims 1 to 9. 11.一种存储软件的计算机可读取介质，其特征在于，所述软件包括能通过一个或多个计算机执行的指令，所述指令在被所述一个或多个计算机执行时执行如所述权利要求1-9中任意一项所述方法的过程。11. A computer-readable medium storing software, characterized in that the software includes instructions that can be executed by one or more computers, and when the instructions are executed by the one or more computers, the process of the method described in any one of claims 1-9 is performed. 12.一种计算机程序产品，包括计算机程序，其特征在于，所述计算机程序被处理器执行时实现权利要求1至9中任一项所述的方法的步骤。12. A computer program product, comprising a computer program, characterized in that when the computer program is executed by a processor, the steps of the method according to any one of claims 1 to 9 are implemented.
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