High-order and low-order mixed inverse finite element deformation reconstruction method based on distributed strain

Abstract

The invention discloses a high-order and low-order mixed inverse finite element deformation reconstruction method based on distributed strain, and belongs to the technical field of finite element deformation reconstruction, and the method comprises the steps: constructing a high-order shell unit in combination with a planar film unit and a plate bending unit; and constructing a high-order inverse shell unit of a complex continuous strain area, constructing a step type inverse shell unit of a cross rib / rib area, and performing boundary transition processing to complete deformation reconstruction. According to the method, the inverse shell unit is constructed in combination with the diaphragm plate unit, and the ribs and discontinuous areas are emphatically processed, so that inverse finite element deformation reconstruction is effectively realized; the boundary is subjected to transition processing, so that the efficiency of inverse finite element deformation reconstruction is improved; and technical support is provided for safety evaluation, failure prevention and control, performance control and even optimization design of the aviation composite material structure.

Description

Technical Field

[0001] The present invention discloses a high - order and low - order hybrid inverse finite - element deformation reconstruction method based on distributed strain, belonging to the technical field of finite - element deformation reconstruction. Background Technique

[0002] In the existing technology related to inverse finite - element shape reconstruction, for relatively complex engineering structures, most of them only conduct accuracy verification based on simulation results. In the content including experimental verification, simple structural forms (such as plates, beams, etc.) and relatively ideal experimental conditions are mostly adopted. At the positions of boundaries, connections, and even defects of real composite structures, their deformations are more complex than those of ideal models, and need to be described by constructing new inverse elements (such as high - order elements). In fact, the inverse finite - element method is based on the finite - element theory and has a broad design space. However, currently, the types of inverse finite - element cells are much less than those of finite - element cells. The reasons come from two aspects: (1) Online shape recognition has relatively high requirements for reconstruction efficiency, which makes it difficult to increase the grid - division density of inverse finite - elements too much, otherwise the problem that the reconstruction speed lags behind the structural deformation speed will occur. This requirement reduces the flexibility of grid meshing; (2) The types of inverse cells are restricted by sensing technology. Taking the element order as an example, although constructing high - order inverse elements can capture the complex deformation characteristics of elements more accurately, existing discrete sensors often cannot provide the number of measurement points required by high - order elements (most existing inverse elements are based on single - point strain measurement), which makes high - order inverse elements unable to play their advantages.

[0003] Distributed optical fibers can provide a much larger measurement - point density than traditional sensors, obtain sufficient strain information of measurement points inside the element, and thus make the construction of high - order inverse elements practically meaningful. Although a large amount of strain data can be provided for inverse elements, the "linear strain measurement" of distributed optical fibers has the dual characteristics of "linear strain measurement" and "one - dimensional linear measurement path". This brings several aspects of influence: (1) Measuring shear strain with optical fibers requires constructing a three - dimensional measurement path (strain rosette) through methods such as optical - fiber bending, which has a certain implementation difficulty. There is an order - of - magnitude difference in the measurement - point density between the direction of the optical - fiber measurement path and its perpendicular direction. At the same time, restricted by factors such as structural working conditions and laying technology, optical - fiber measurement points cannot achieve large - area coverage in two - dimensional space. The above factors result in only the deformation of some inverse elements or only some deformation components inside the element being measurable. In addition, distributed optical fibers often have necessary curved paths, and there are difficulties in making full use of the line - strain measurement data with variable directions in inverse finite - elements; (2) Different from the grid meshing of traditional finite - elements, inverse finite - elements need to synchronously plan grid meshing and sensing paths. When the grid changes, the measurement path often needs to be adjusted synchronously. If the grid is too complex, the entire planning process will be very cumbersome. (3)

[0004] The essence of the above problems is the co - design problem of fiber - optic measurement parameters and inverse element parameters, which brings about the trade - off problem of the size and shape of inverse elements. If the elements are too small (to adapt to the density of path directions), it will lead to a decrease in computational efficiency and there will be a large number of "empty elements" without measured point values. If the elements are too large, there will be a problem of insufficient data utilization. Summary of the Invention

[0005] The object of the present invention is to provide a high - order and low - order hybrid inverse finite - element deformation reconstruction method based on distributed strain to solve the problem in the prior art that it is impossible to find suitable elements in the inverse finite - element shape reconstruction, resulting in low computational efficiency or insufficient data utilization.

[0006] The high - order and low - order hybrid inverse finite - element deformation reconstruction method based on distributed strain includes:

[0007] S1. Combine plane membrane elements and plate bending elements to construct high - order shell elements;

[0008] S2. Construct high - order inverse shell elements for complex continuous strain regions;

[0009] S3. Construct step - type inverse shell elements for cross - rib / rib regions;

[0010] S4. Perform boundary transition processing to complete deformation reconstruction.

[0011] In the high - order shell elements, x, y, and z are used as the three - dimensional coordinate directions. In the plate bending elements, θ x , θ y , and w are used as the three - dimensional coordinate directions. In the plane membrane elements, u and v are used as the two - dimensional coordinate directions.

[0012] In the plane membrane elements, for a 9 - node quadrilateral element, taking the 9th node as the origin of the natural coordinates of the element, the expression form of the shape function is:

[0013] N1=(1 + r)(1 + s) / 4−N9 / 4;

[0014] N2=(1 - r)(1 + s) / 4−N9 / 4;

[0015] N3=(1 - r)(1 - s) / 4−N9 / 4;

[0016] N4=(1 + r)(1 - s) / 4−N9 / 4;

[0017] N5=(1 - r 2 )(1 + s) / 2−N9 / 2;

[0018] N6=(1 - r)(1 - s 2 ) / 2−N9 / 2;

[0019] N7 = (1 - r 2 )(1 - s) / 2 - N9 / 2;

[0020] N8 = (1 + r)(1 - s 2 ) / 2 - N9 / 2;

[0021] N9 = (1 - r 2 )(1 - s 2 );

[0022] Wherein, N1, N2, N3, N4, N5, N6, N7, N8, N9 are the shape functions of 9 nodes respectively, r and s are the longitudinal displacement and transverse displacement in the natural coordinates of the element respectively. Regarding r and s as the two-dimensional coordinate directions of each node, the two-dimensional coordinate direction is equivalent to the two translational degrees of freedom u and v:

[0023]

[0024] Wherein, i represents the i-th node, N i represents the shape function of the i-th node, u i , v i represent the translational degrees of freedom of the i-th node respectively, and the membrane strain e(u e ) of the plane membrane is:

[0025]

[0026] B m = [B m1 B m2 B m3 B m4 B m5 B m6 B m7 B m8 ;

[0027] Wherein, is the displacement vector of the shell element, w i is the flexural degree of freedom of the i-th node, θ xi and B yi are the two in-plane rotation degrees of freedom of the i-th node, θ zi is the out-of-plane rotation degree of freedom of the i-th node, B m is the node displacement vector, B m1 , B m2 , B m3 , B m4 , B m5 , B m6 , B m7 , B m8 are the node displacements corresponding to N1, N2, N3, N4, N5, N6, N7, N8.

[0028] In the plate bending element, N9 at the center has no flexural degree of freedom, and the nodal displacement vector of the plate bending element is:

[0029]

[0030] At the position with the thickness coordinate value of z, the translational degree of freedom is:

[0031]

[0032] The in-plane strain state ε1 of the plate bending element is:

[0033]

[0034] B κ = [B κ1 B κ2 B κ3 B κ4 B κ5 B κ6 B κ7 B κ8 B κ9 ;

[0035] Wherein, B κ is the curvature interpolation matrix, and B κ1 , B κ2 , B κ3 , B κ4 , B κ5 , B κ6 , B κ7 , B κ8 , B κ9 are the curvature interpolations of N1, N2, N3, N4, N5, N6, N7, N8, N9;

[0036] The shear strain state ε2 is:

[0037]

[0038] B g = [B g1 B g2 B g3 B g4 B g5 B g6 B g7 B g8 B g9 ;

[0039] Wherein, B g is the transverse shear strain interpolation matrix, and B 91 B g2 Bg3 B g4 B g5 B g6 B g7 B 98 B g9 is the interpolation of the transverse shear strains of N1, N2, N3, N4, N5, N6, N7, N8, N9.

[0040] B m , B κ , B g The matrix elements in are respectively:

[0041]

[0042] In the formula, N i,x , N i,y are respectively the x and y direction components of N i .

[0043] The plane membrane element and the plate bending element are directly superposed at the element level, and then the coordinates of the plane membrane element and the plate bending element are transformed into the coordinates of the high-order shell element.

[0044] The strain vector ε of the high-order shell element is:

[0045]

[0046] The error functional φ e (u e ) is:

[0047] φ e (u e ) = w e ‖e(u e ) - e ε ‖ 2 + w κ ‖κ(u e ) - κ ε ‖ 2 + w g ‖g(u e ) - g ε ‖ 2 ;

[0048]

[0049] In the formula, e(u e ), κ(u e ), g(u e ) are the three theoretical strain values of the high-order shell element, w e , w κ , w gis the weight of three strain values, e ε , κ ε , g ε are the strain measurement values of three strain values.

[0050] Constructing a high - order inverse shell element for a complex continuous strain region includes taking the first - order variation of the error functional of the high - order shell element to obtain the solution equation of the inverse finite element:

[0051]

[0052] In the formula, k e is the element coefficient matrix, f e is the load vector of the element, A e is the integral limit value, h is an adjustable parameter, and the solution equation of the inverse finite element is obtained:

[0053] KU = F;

[0054]

[0055] In the formula, nel is the number of elements, K is the total system coefficient matrix, U is the overall displacement vector of the structure, F is the overall load vector, T e is the coordinate transformation matrix.

[0056] Constructing a step - type inverse shell element for the cross - rib / stringer region includes that the stress - strain relationship per unit area of the composite laminate is:

[0057] N = Ae;

[0058] In the formula, A is the tensile stiffness, N is the stress of the composite laminate, e is the strain of the composite laminate. Introduce the Heaviside unit step function H i , when the independent variable is negative, H i is equal to 0, when the independent variable is positive, H i is equal to 1, and the stiffness distribution is expressed as:

[0059]

[0060] In the formula, A0 is the tensile stiffness matrix at the non - stiffened position, ΔA is the difference between the tensile stiffness matrices at the stiffened and non - stiffened positions, which is equivalent to the tensile stiffness matrix corresponding to the ribs;

[0061] For the non - stiffened structure, the strain distribution e0 inside the element under the action of internal forces is:

[0062]

[0063] For the stiffened structure, under the same internal force load condition, the strain distribution inside the element is:

[0064] e = A -1 N;

[0065] The strain distribution relationship between the reinforced unit and the unreinforced unit under the action of internal force load is:

[0066] e = A -1 A0e0 = (L A A0) -1 A0e0.

[0067] Boundary transition processing includes processing discontinuous strain and processing unit transition;

[0068] Processing discontinuous strain includes using the following interpolation function f(x, y) to represent the in-plane displacement field:

[0069] f(x, y) = Y(y)X(x);

[0070] X(x) = [N1, N2, N3, N4];

[0071] N1 = 1 + f1(x)α1α;

[0072] N2 = x - f1(x)α5α - f2(x)α2α;

[0073] N3 = -f1(x)α2α - f2(x)α1α;

[0074] N4 = f1(x)α4α - f2(x)α3α;

[0075]

[0076] In the formula, r i is the inradius of the shell element;

[0077] Processing unit transition includes that there are coordinated element assemblies and uncoordinated element assemblies between low-order elements and high-order elements. For the coordinated element assemblies, when calculating, reintroduce the 5-node element, and use the inverse finite element method to synthesize the overall stiffness matrix and load matrix for calculation;

[0078] For the uncoordinated element assemblies, the transition boundary field function is uncoordinated, and constraint equations are introduced to ensure coordination:

[0079]

[0080] In the formula, is the constraint term, is the transition boundary field function, at the transition boundary σ = 1, and at other places σ = 1×10 -5 .

[0081] Compared with the prior art, the present invention has the following beneficial effects: The present invention combines the template unit to construct the inverse shell unit, and focuses on the treatment of ribs and discontinuous regions, effectively realizing the inverse finite element deformation reconstruction; performs transitional treatment on the boundary, improving the efficiency of inverse finite element deformation reconstruction; provides technical support for the safety assessment, failure prevention and control, performance control, and even optimization design of aerospace composite material structures. Description of the Drawings

[0082] Figure 1 is the template unit laminated to the shell unit;

[0083] Figure 2 is a high-order 9-node shell unit;

[0084] Figure 3 is a planar membrane unit with 9 nodes;

[0085] Figure 4 is a plate bending unit with 9 nodes;

[0086] Figure 5 is a compatible boundary transition unit;

[0087] Figure 6 is a non-compatible boundary transition unit;

[0088] Figure 7 is a combination scheme of compatible boundary transition units;

[0089] Figure 8 is a combination scheme of non-compatible boundary transition units. Detailed Embodiments

[0090] To make the objectives, technical solutions, and advantages of the present invention clearer, the technical solutions in the present invention are clearly and completely described below. Obviously, the described embodiments are part of the embodiments of the present invention, rather than all of the embodiments. Based on the embodiments in the present invention, all other embodiments obtained by those of ordinary skill in the art without creative efforts belong to the scope of protection of the present invention.

[0091] The high-low order hybrid inverse finite element deformation reconstruction method based on distributed strain includes:

[0092] S1. Combine the planar membrane unit and the plate bending unit to construct a high-order shell unit;

[0093] S2. Construct a high-order inverse shell unit for complex continuous strain regions;

[0094] S3. Construct a step-type inverse shell unit for the cross-rib / rib region;

[0095] S4. Perform boundary transition treatment to complete the deformation reconstruction.

[0096] In high-order shell elements, x, y, and z are used as the three-dimensional coordinate directions. In plate bending elements, θ x , θ y , and w are used as the three-dimensional coordinate directions. In plane membrane elements, u and v are used as the two-dimensional coordinate directions.

[0097] In plane membrane elements, for a 9-node quadrilateral element, with the 9th node as the origin of the natural coordinates of the element, the expression form of the shape function is:

[0098] N1 = (1 + r)(1 + s) / 4 - N9 / 4;

[0099] N2 = (1 - r)(1 + s) / 4 - N9 / 4;

[0100] N3 = (1 - r)(1 - s) / 4 - N9 / 4;

[0101] N4 = (1 + r)(1 - s) / 4 - N9 / 4;

[0102] N5 = (1 - r 2 )(1 + s) / 2 - N9 / 2;

[0103] N6 = (1 - r)(1 - s 2 / 2 - N9 / 2;

[0104] N7 = (1 - r 2 )(1 - s) / 2 - N9 / 2;

[0105] N8 = (1 + r)(1 - s 2 ) / 2 - N9 / 2;

[0106] N9 = (1 - r 2 )(1 - s 2 );

[0107] In the formula, N1, N2, N3, N4, N5, N6, N7, N8, and N9 are the shape functions of the 9 nodes respectively. r and s are the longitudinal displacement and transverse displacement in the natural coordinates of the element respectively. Regarding r and s as the two-dimensional coordinate directions of each node, the two-dimensional coordinate directions are equivalent to the two translational degrees of freedom u and v:

[0108]

[0109] In the formula, i represents the i-th node, N i represents the shape function of the i-th node, u i , v i respectively represent the translational degrees of freedom of the i-th node. The membrane strain e(u e ) of the plane membrane is:

[0110]

[0111] B m = [B m1 B m2 B m3 B m4 B m5 B m6 B m7 B m8 ;

[0112] In the formula, is the displacement vector of the shell element, and w i is the flexural degree of freedom of the i-th node, and θ xi and θ yi are the two in-plane rotation degrees of freedom of the i-th node, and θ zi is the out-of-plane rotation degree of freedom of the i-th node, B m is the node displacement vector, B m1 , B m2 , B m3 , B m4 , B m5 , B m6 , B m7 , B m8 are the node displacements corresponding to N1, N2, N3, N4, N5, N6, N7, N8.

[0113] In the plate bending element, N9 at the center has no flexural degree of freedom, and the node displacement vector of the plate bending element is:

[0114]

[0115] At the position with the thickness coordinate value of z, the translational degree of freedom is:

[0116]

[0117] The in-plane strain state ε1 of the plate bending element is:

[0118]

[0119] B κ = [B κ1 B κ2 B κ3 B κ4 B κ5 B κ6 B κ7 B κ8 B κ9 ;

[0120] In the formula, B κis the curvature interpolation matrix, B κ1 and B κ2 and B κ3 and B κ4 and B κ5 and B κ6 and B κ7 and B κ8 and B κ9 are the curvature interpolations of N1, N2, N3, N4, N5, N6, N7, N8, N9;

[0121] The shear strain state ε2 is:

[0122]

[0123] B g = [B g1 B g2 B g3 B g4 B g5 B g6 B g7 B g8 B g9 ;

[0124] In the formula, B g is the transverse shear strain interpolation matrix, and B g1 B g2 B g3 B g4 B g5 B g6 B g7 B g8 B g9 are the transverse shear strain interpolations of N1, N2, N3, N4, N5, N6, N7, N8, N9.

[0125] B m and B κ and B g The matrix elements in are respectively:

[0126]

[0127] In the formula, N i,x and N i,y are respectively the x and y direction components of N i .

[0128] The plane membrane element and the plate bending element are directly superposed at the element level, and then the coordinates of the plane membrane element and the plate bending element are transformed into the coordinates of the high-order shell element.

[0129] The strain vector ε of the high-order shell element is:

[0130]

[0131] Error functional Φ of the high-order shell element e (u e ) is as follows:

[0132] Φ e (u e ) = w e ‖e(u e ) - e ε ‖ 2 + w κ ‖κ(u e ) - κ ε ‖ 2 + w g ‖g(u e ) - g ε ‖ 2 ;

[0133]

[0134] In the formula, e(u e ), κ(u e ), g(u e ) are the three theoretical strain values of the high-order shell element, w e , w κ , w g are the weights of the three strain values, and e ε , κ ε , g ε are the measured strain values of the three strain values.

[0135] Constructing the high-order inverse shell element for a complex continuous strain region includes taking the first variation of the error functional of the high-order shell element to obtain the solution equation of the inverse finite element:

[0136]

[0137] In the formula, k e is the element coefficient matrix, f e is the element load vector, A e is the integration limit value, and h is an adjustable parameter, to obtain the inverse finite element solution equation:

[0138] KU = F;

[0139]

[0140] In the formula, nel is the number of elements, K is the total system coefficient matrix, U is the overall displacement vector of the structure, F is the overall load vector, and T e is the coordinate transformation matrix.

[0141] The stepped inverse shell element for constructing the cross - rib region includes that the stress - strain relationship per unit area of the composite laminate is:

[0142] N = Ae;

[0143] where A is the tensile stiffness, N is the stress of the composite laminate, e is the strain of the composite laminate, and the Heaviside unit step function H i is introduced. When the independent variable is negative, H i equals 0. When the independent variable is positive, H i equals 1. The stiffness distribution is expressed as:

[0144]

[0145] where A0 is the tensile stiffness matrix at the non - stiffened position, and ΔA is the difference between the tensile stiffness matrices at the stiffened and non - stiffened positions, which is equivalent to the tensile stiffness matrix corresponding to the rib;

[0146] For the non - stiffened structure, the strain distribution e0 inside the element under the action of internal forces is:

[0147]

[0148] For the stiffened structure, under the action of the same internal force load, the strain distribution inside the element is:

[0149] e = A -1 N;

[0150] The strain distribution relationship between the stiffened element and the non - stiffened element under the action of internal force load is:

[0151] e = A -1 A0e0=(L A A0) -1 A0e0.

[0152] The boundary transition treatment includes dealing with discontinuous strains and dealing with element transitions;

[0153] Dealing with discontinuous strains includes using the following interpolation function f(x, y) to represent the in - plane displacement field:

[0154] f(x, y)=Y(y)X(x);

[0155] X(x)=[N1, N2, N3, N4];

[0156] N1 = 1 + f1(x)α1α;

[0157] N2 = x - f1(x)α5α - f2(x)α2α;

[0158] N3 = -f1(x)α2α - f2(x)α1α;

[0159] N4 = f1(x)α4α - f2(x)α3α;

[0160]

[0161] In the formula, r i is the in - circle radius of the shell element;

[0162] The transition of the processing unit includes that there are coordinated unit combinations and uncoordinated unit combinations between low - order units and high - order units. For the coordinated unit combination, when calculating, a 5 - node unit is re - introduced, and the inverse finite element method is used to synthesize the overall stiffness matrix and load matrix for calculation;

[0163] For the uncoordinated unit combination, the transition boundary field function is uncoordinated, and constraint equations are introduced to ensure coordination:

[0164]

[0165] In the formula, is the constraint term, is the transition boundary field function, where σ = 1 at the transition boundary and σ = 1×10 -5 .

[0166] The present invention aims at the major requirements of the service safety and service performance of aerospace structures, with the goal of improving the shape perception accuracy and efficiency of composite thin - wall structures, and conducts research on the inverse finite element shape reconstruction method based on high - density linear strain measurement. Aiming at the problem of low shape reconstruction accuracy in complex strain regions, construction methods of high - order inverse shell elements and step - type inverse shell elements in the rib / stringer region are respectively proposed, and an inverse shell element matrix transformation and assembly method based on linear strain measurement in any direction is developed; aiming at the problem of balancing the shape reconstruction accuracy and efficiency of iFEM, a high - convenience mesh generation scheme and a hybrid application method of high / low - order inverse shell elements are proposed, a sensor path planning method for different numerical integration schemes is developed, and an element - level adaptive weighting strategy based on the inverse finite element functional formulation is proposed; aiming at the influence of multi - source uncertainties in real structures on the reconstruction accuracy, a quantification and control model of uncertain factors is established, and an iterative correction method for the inverse finite element model is proposed. Finally, bending - torsion coupling and cyclic loading tests of an aerospace composite thin - wall stiffened box - section structure are designed and carried out to verify the accuracy and efficiency of the method. The project results provide technical support for the safety assessment, failure prevention and control, performance control, and even optimization design of aerospace composite structures.

[0167] In the construction of high-order inverse shell elements for complex continuous strain regions, for the boundary regions, low-order elements cannot adapt to large changes in strain gradients, resulting in an increase in computational errors. However, increasing the number of elements will lead to a decrease in computational efficiency and cannot meet the requirements of reconstruction problems. Therefore, large high-order elements need to be used to calculate this region. When constructing shell elements, the mid-plane plane stress membrane element and the plate bending element are directly superimposed at the element level. Subsequently, after the membrane-bending coupling is assembled during the transformation from the local coordinate system to the global coordinate system, the construction of the shell element can be achieved, as shown in Figure 1 shown. According to the element superposition idea, arbitrarily many-node high-order elements can be constructed. For the MITS9 (Mixed Interpolated Tensorial Shell with 9 nodes) shell element shown in Figure 2 , the element can perform plane stretching and transverse bending. In the mid-plane membrane element, the nodes will have translational degrees of freedom u i and v i . In the shell bending element, the nodes will have flexural degrees of freedom and in-plane rotational degrees of freedom w i , θ xi and θ yi . To perform the subsequent transformation from the local coordinate system to the global coordinate system, the out-of-plane rotational degree of freedom θ zi is added, making θ zi ≡ 0 to form the displacement vector expression of the element.

[0168] The plane membrane element with 9 nodes is as shown in Figure 3 , and the plate bending element with 9 nodes is as shown in Figure 4 . In the MITC9 element, the 8 nodes on the boundary have degrees of freedom w i , θ xi and θ yi . Due to the coupling relationship, the central 9 nodes only have independent degrees of freedom θ xi and θ yi , that is, there is no independent degree of freedom w9, and the corresponding strain matrix coefficient is 0. According to the thin film analogy theory, the shape functions of the plate bending element can still be expressed by the shape functions of the same-node membrane element.

[0169] Since the box section only bears in-plane loads, strain sensors can be pasted only on the upper surface of the structure to measure the mid-plane strain of the structure. There are two existing measurement schemes for the measurement of discontinuous strain. Scheme 1: Paste sensors at positions without clamping bars (where the strain distribution is not affected by the ribs), and then calculate using the formula derived in the previous section. Scheme 2: Directly paste distributed optical fibers on the upper surface of the structure to measure the strain distribution of the structure, and then extract the corresponding strain data according to requirements. For 8-node high-order elements, numerical calculations can be carried out using the Gaussian integration formula. To ensure the accuracy of the results, a 3×3 point Gaussian integration scheme is adopted. The advantage of high-order elements is that larger elements can be set for calculation, and the complex strain region can meet the accuracy requirements without using dense mesh division. On the premise of meeting the bending radius of the distributed optical fiber, one distributed optical fiber sensor can be used to measure the strain data of 9 Gaussian integration points, meeting the requirements of deformation reconstruction.

[0170] To further improve the calculation efficiency, low-order elements with higher calculation efficiency are applied in areas with smaller strain gradients, and high-order elements are used in areas with larger strain gradients. The mixing of the two types of elements will inevitably result in element transitions. At the same time, to maintain grid consistency and ensure the same grid size, there will be two types of element transition situations. Coordinated and uncoordinated two-dimensional element assemblies, such as Figure 5 and Figure 6 shown. In the coordinated element assembly, the coordinates and field functions of the element transition boundary both change quadratically, and the number of interpolation nodes is the same. A 5-node element is used for the connection between 4-node elements and 8-node elements in the coordinated transition, as shown in Figure 7 and Figure 8 shown. During the calculation, only the 5-node element needs to be reintroduced, and the overall stiffness matrix and load matrix can be synthesized using the inverse finite element method for calculation.

[0171] The above embodiments are only used to illustrate the technical solutions of the present invention, rather than to limit it. Although the present invention has been described in detail with reference to the foregoing embodiments, those of ordinary skill in the art should understand that: they can still modify the technical solutions recorded in the foregoing embodiments, or perform equivalent replacements on some or all of the technical features, and these modifications or replacements do not cause the essence of the corresponding technical solutions to deviate from the scope of the technical solutions of the embodiments of the present invention.

Claims

1. A high- and low-order mixed inverse finite element deformation reconstruction method based on distributed strain, characterized in that: include: S1. Combine the plane membrane element and the plate bending element to construct a high-order shell element; S2, construct high-order inverse shell elements in complex continuous strain regions; S3, construct the stepped inverse shell element across the rib / rib region; S4. Perform boundary transition processing to complete deformation reconstruction.

2. The high-low order mixed inverse finite element deformation reconstruction method based on distributed strain according to claim 1 is characterized in that: In high-order shell elements, x, y, and z are used as the three-dimensional coordinate directions, and in plate bending elements, θ is used. x ,θ y , w are used as the three-dimensional coordinate directions; in the plane membrane unit, u, v are used as the two-dimensional coordinate directions.

3. The high-low order mixed inverse finite element deformation reconstruction method based on distributed strain according to claim 2 is characterized in that: In the plane membrane element, for a 9-node quadrilateral element, the 9th node is taken as the origin of the element's natural coordinates, and the shape function is expressed as: N1=(1+r)(1+s) / 4-N9 / 4; N2=(1-r)(1+s) / 4-N9 / 4; N3=(1-r)(1-s) / 4-N9 / 4; N4=(1+r)(1-s) / 4-N9 / 4; N5=(1-r 2 )(1+s) / 2-N9 / 2; N6=(1-r)(1-s 2 ) / 2-N9 / 2; N7=(1-r 2 )(1-s) / 2-N9 / 2; <h2 style=";text-align:left;direction:ltr">N8 = (1+r)(1-s)<h2 style=";text-align:left;direction:ltr"> 2 <h2 style=";text-align:left;direction:ltr"> ) / 2-N9 / 2; N9=(1-r 2 )(1-s 2 ); Where N1, N2, N3, N4, N5, N6, N7, N8, and N9 are the shape functions of the nine nodes, r and s are the longitudinal displacement and lateral displacement in the natural coordinates of the unit, respectively. r and s are regarded as the two-dimensional coordinate directions of each node, which are equivalent to the two translational degrees of freedom u and v: In the formula, i represents the i-th node, N i represents the shape function of the ith node, u i 、v i denote the translational freedom of the ith node, the membrane strain e(u e )for: B m =[B m1 B m2 B m3 B m4 B m5 B m6 B m7 B m8 ]; In the formula, is the shell element displacement vector, w i is the flexural degree of freedom of the ith node, θ xi and θ yi are the two in-plane rotational degrees of freedom of the ith node, θ zi is the out-of-plane rotational degree of freedom of the ith node, B m is the node displacement vector, B m1 , B m2 , B m3 , B m4 , B m5 , B m6 , B m7 , B m8 are the node displacements corresponding to N1, N2, N3, N4, N5, N6, N7, and N8.

4. The high-low order mixed inverse finite element deformation reconstruction method based on distributed strain according to claim 3 is characterized in that: In the plate bending unit, N9 located in the center does not have flexural freedom, and the node displacement vector of the plate bending unit is: At the thickness coordinate value z, the translational degrees of freedom are: The in-plane strain state ε1 of the plate bending element is: B κ =[B κ1 B κ2 B κ3 B κ4 B κ5 B κ6 B κ7 B κ8 B κ9 ]; In the formula, B κ is the curvature interpolation matrix, B κ1 , B κ2 , B κ3 , B k4 , B k5 , B k6 , B κ7 , B k8 , B k9 is the curvature interpolation of N1, N2, N3, N4, N5, N6, N7, N8, and N9; The shear strain state ε2 is: B g =[B g1 B g2 B g3 B g4 B g5 B g6 B g7 B g8 B g9 ]; In the formula, B g is the transverse shear strain interpolation matrix, B g1 B g2 B g3 B g4 B g5 B g6 B g7 B g8 B g9 It is the interpolated transverse shear strain of N1, N2, N3, N4, N5, N6, N7, N8 and N9.

5. The high-low order mixed inverse finite element deformation reconstruction method based on distributed strain according to claim 4 is characterized in that: B m , B κ , B g The matrix elements in are: Where N i,x 、N i,y N i The x and y components of .

6. The high-low order mixed inverse finite element deformation reconstruction method based on distributed strain according to claim 5 is characterized in that: The plane membrane unit and the plate bending unit are directly superimposed on the unit level, and then the coordinates of the plane membrane unit and the plate bending unit are converted to the high-order shell unit coordinates.

7. The high-low order mixed inverse finite element deformation reconstruction method based on distributed strain according to claim 6 is characterized in that: The strain vector ε of the high-order shell element is: Error functional Φ for higher-order shell elements e (u e )for: Φ e (u e )jw e ‖e(u e )-e ε ‖ 2 +w κ ‖κ(u e )-κ ε ‖ 2 +w g ‖g(u e )-g ε ‖ 2 100. In the formula, e(u e ),κ(u e )、g(u e ) are the three strain theoretical values ​​of the high-order shell element, w e 、w κ 、w g are the weights of the three strain values, e ε , κ ε , g ε Strain measurements for three strain values.

8. The high-low order mixed inverse finite element deformation reconstruction method based on distributed strain according to claim 7 is characterized in that: Constructing high-order inverse shell elements in complex continuous strain regions includes taking the first-order variation of the error functional of the high-order shell element to obtain the solution equation of the inverse finite element: In the formula, k e is the unit coefficient matrix, f e is the load vector of the element, A e is the integral limit, h is an adjustable parameter, and the inverse finite element solution equation is obtained: KU=F; Where nel is the number of elements, K is the overall coefficient matrix, U is the overall displacement vector of the structure, F is the overall load vector, and T e is the coordinate transformation matrix.

9. The high-low order mixed inverse finite element deformation reconstruction method based on distributed strain according to claim 8 is characterized in that: The stepped inverse shell element for the cross-rib region includes: The stress-strain relationship per unit area of ​​the composite laminate is: N = Ae; In the formula, A is the tensile stiffness, N is the stress of the composite laminate, e is the strain of the composite laminate, and the Heaviside unit step function H is introduced. i , when the independent variable is negative, H i is equal to 0. When the independent variable is positive, H i Equal to 1, the stiffness distribution is expressed as: Where A0 is the tensile stiffness matrix at the unreinforced position, ΔA is the difference between the tensile stiffness matrices at the reinforced and unreinforced positions, which is equivalent to the tensile stiffness matrix corresponding to the ribs; For unreinforced structures, the strain distribution e0 inside the unit under the condition of internal force is: For the reinforced structure, the strain distribution inside the unit under the same internal force load condition is: and=A -1 No; The strain distribution relationship between reinforced units and unreinforced units under internal force load conditions is: e=A -1 A0e0=(L A A0) -1 A0e0.

10. The high-low order mixed inverse finite element deformation reconstruction method based on distributed strain according to claim 9 is characterized in that: Boundary transition processing includes processing discontinuous strain and processing unit transition; The treatment of discontinuous strains involves representing the displacement field within the surface using the following interpolation function N(x,y): f(x,y)=Y(y)X(x); X(x) = [N1, N2, N3, N4]; N1=1+f1(x)α1α; N2=x-f1(x)α5α-f2(x)α2α; N3=-f1(x)α2α-f2(x)α1α; N4=f1(x)α4α-f2(x)α3α; In the formula, r i is the radius of the shell element inscribed circle; The processing unit transition includes the existence of coordinated unit combinations and non-coordinated unit combinations between low-order units and high-order units. For the coordinated unit combinations, the 5-node unit is reintroduced during calculation, and the overall stiffness matrix and load matrix are synthesized using the inverse finite element method for calculation; For the combination of non-coordinated units, the transition boundary field function is not coordinated, and the constraint equation is introduced to ensure the coordination: In the formula, is a constraint, is the transition boundary field function, where σ = 1 at the transition boundary and σ = 1×10 -5 .

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