A displacement reconstruction method for arbitrary boundary floating raft structure based on improved inverse finite element method

Abstract

The present application relates to a kind of displacement reconstruction method based on improved inverse finite element method suitable for arbitrary boundary floating raft structure, the present application constructs unified constraint least square mathematical framework fusing fixed boundary and elastic boundary, based on displacement measurement data, angle measurement data, strain measurement data, the constraint condition updating of mathematical framework is carried out, based on elimination method, it is proposed that arbitrary boundary inverse finite element method develops displacement field solution, effectively expands the universality of existing inverse finite element method to special boundary constraint (such as elastic support). Compared with prior art, the present application fuses strain data and displacement data, effectively solves the displacement reconstruction problem of floating raft structure under the constraint of arbitrary boundary condition, can be widely popularized in the field such as ship, aerospace and other fields facing large raft structure deformation needs to carry out deformation reconstruction.

Description

Technical Field

[0001] This invention relates to the field of floating raft displacement reconstruction methods, and specifically to a displacement reconstruction method based on an improved inverse finite element method applicable to floating raft structures with arbitrary boundaries. Background Technology

[0002] Floating raft vibration isolation technology is widely used in ship propulsion systems due to its excellent vibration isolation performance and compact structure. The floating raft is connected to the hull via elastic elements (such as airbag isolators), constrained by elasticity. The attitude of the floating raft affects the relative position of the equipment on board and the alignment of the shafting, thus impacting equipment operation. Especially in propulsion systems, a good shafting alignment is a prerequisite for effective vibration isolation. National military standards and overall ship technical requirements stipulate that after the main engine is elastically installed, the shafting offset should be ≤0.2~0.5mm, and the skewness ≤0.5mm / m. Poor shafting alignment will jeopardize operational safety, increase vibration noise, and severe misalignment may even endanger the safe operation of the propulsion system. As floating rafts become larger, more integrated, and lighter, their span dimensions are constantly increasing, inevitably leading to a decrease in stiffness and a tendency to generate large non-uniform elastic deformations. The inverse finite element method, based on variational principles, constructs a weighted residual functional of the measured strain and theoretical strain fields, and reconstructs the displacement field by minimizing this functional. For displacement reconstruction of structural deformation, the currently known inverse finite element deformation reconstruction technology (patent numbers 202311535664.3, 202310494306.6, 202211128284.3, 202210312365.2, 202210019848.3, 202110806838.X, 202011615128.0) has significant theoretical limitations when dealing with non-fixed boundary conditions: its mathematical framework can only rigorously characterize ideal fixed boundaries with zero displacement / zero rotation, thus limiting its applicability to displacement reconstruction of structures with elastic boundaries. This paper aims to study a structural displacement reconstruction method applicable to both fixed and elastic boundaries, particularly elastic boundaries. Summary of the Invention

[0003] To address the aforementioned issues, this invention proposes an improved inverse finite element method applicable to displacement reconstruction of raft structures with arbitrary boundaries. This method adds constraints to the displacement data, constructs a mathematical model for reconstructing the displacement field of arbitrary boundaries that satisfies both fixed and elastic boundaries, and uses the elimination method to solve the problem, thus establishing the improved inverse finite element method for reconstructing displacement of structures with arbitrary boundaries.

[0004] The present invention adopts the following technical solution:

[0005] A displacement reconstruction method based on the improved inverse finite element method, applicable to floating raft structures with arbitrary boundaries, includes the following steps:

[0006] Step 1: Under the condition of elastic boundary, determine the specifications, quantity and position of the elastic elements on the solid floating raft, arrange strain measuring points and displacement measuring points on the upper and lower surfaces of the solid floating raft, apply loads at designated nodes of the solid floating raft, measure the strain and displacement of the solid floating raft, and obtain the strain measurement data and displacement measurement data of the solid floating raft.

[0007] Under fixed boundary conditions, strain measurement points are arranged on the upper and lower surfaces of the solid floating raft, and loads are applied at designated nodes of the solid floating raft to measure the strain of the solid floating raft and obtain strain measurement data of the solid floating raft.

[0008] Step 2: Calculate the global stiffness matrix based on the strain measurement data of the solid floating raft. and class load vector ;

[0009] Step 3: Combining measured displacement and strain data, the constrained linear least squares problem under arbitrary boundary conditions is obtained:

[0010]

[0011] In the formula This represents the displacement vector to be determined. express The estimated value, Given the location Boolean matrix of the measured displacement data, under elastic boundary conditions, For actual displacement measurement data, under fixed boundary conditions, =0;

[0012] Step 4: Solve the above constrained least squares problem using numerical methods to obtain the optimal solution U. This is the full-field reconstruction displacement.

[0013] Furthermore, step 1 specifically includes the following steps:

[0014] A grid is divided on the upper and lower surfaces of the solid floating raft, and strain and displacement measuring points are arranged at the nodes of the grid. The number of strain measuring points on the upper and lower surfaces of the solid floating raft is the same and they correspond one-to-one.

[0015] Furthermore, in step 2, the global stiffness matrix... and class load vector It is calculated using the following formula:

[0016]

[0017] In the formula, Indicates membrane strain, Indicates the curvature of the curve. , , Let be the displacement-strain transformation matrices for the corresponding components. , , These are the error weighting coefficients for the membrane strain, bending curvature, and transverse shear strain components, respectively.

[0018] , , , where n is the total number of strain measurement points on the upper or lower surface of the floating raft, and the measured strains at the i-th strain measurement point on the upper or lower surface of the solid floating raft are respectively and Then the membrane strain at the i-th mid-surface and curvature The calculation formula is:

[0019] .

[0020] The beneficial effects of this invention are as follows:

[0021] The improved inverse finite element method of this invention can achieve full-field displacement reconstruction under arbitrary boundary conditions, breaking through the dependence of existing methods on zero boundary conditions.

[0022] Compared with existing technologies, this invention integrates strain data and displacement data, effectively solving the problem of displacement reconstruction of floating raft structures under arbitrary boundary conditions. It can be widely applied in fields such as shipbuilding and aerospace, where large raft structures need to undergo deformation reconstruction.

[0023] The present invention will now be described in detail with reference to the accompanying drawings and embodiments. Attached Figure Description

[0024] Figure 1 This is a flowchart of the method of the present invention.

[0025] Figure 2 This is a schematic diagram of the grid division and sensor arrangement of a floating raft structure provided in an embodiment of the present invention.

[0026] Figure 3 This is a comparison diagram of the reconstructed displacement and the reference displacement of the cantilever structure of this invention. (P3 applies a vertical load)

[0027] Figure 4 This is a comparison diagram of the displacement reconstruction amount and displacement reference amount of the elastic support structure of the present invention (P1 is where the vertical load is applied).

[0028] The attached diagram lists the components represented by each number as follows:

[0029] Among them, N1~N18 are displacement measuring points, S1~S10 are strain measuring points, P1~P4 are vertical weight application points, and spring1~spring6 are elastic support points. A total of 10 four-node inverse elements are defined (e.g., nodes N7-N8-N14-N13 together constitute one inverse element), and the size of the inverse elements is uniformly defined as length. ,width . Detailed Implementation

[0030] The following examples are provided to help you better understand the present invention, but they are not intended to limit the invention.

[0031] A displacement reconstruction method based on the improved inverse finite element method (IFM) applicable to raft structures with arbitrary boundaries is proposed. A unified constraint least-squares mathematical framework integrating fixed and elastic boundaries is constructed. The constraints of the mathematical framework are updated based on displacement, rotation, and strain measurement data. An arbitrary boundary IFM displacement field solution is proposed based on the elimination method, effectively extending the universality of the existing IFM for special boundary constraints (such as elastic supports). The method includes the following steps:

[0032] (1) Divide the structure into elements, derive the weighted least squares error functional of theoretical strain and measured strain using the conventional inverse finite element method, and minimize it to obtain the following model:

[0033]

[0034] In the formula Let be the nodal displacement vector. For global stiffness matrix, For class load vectors, the specific mathematical form is:

[0035]

[0036] In the formula, Indicates membrane strain, Indicates the curvature of the curve. This represents the transverse shear strain, which is directly set to 0. , , Let be the displacement-strain transformation matrices for the corresponding components. , , These are the error weighting coefficients for the membrane strain, bending curvature, and transverse shear strain components, respectively. In the formula... It is only related to mesh generation and structural parameters. It needs to be calculated based on real-time strain data. , , They can be determined based on the upper and lower surfaces of the structure ( ) measured strain and The formula for calculating the strain at the mid-surface is:

[0037]

[0038] (2) Combining measured displacement and strain data, the constrained linear least squares problem under arbitrary boundary conditions is obtained:

[0039]

[0040] In the formula Given a Boolean matrix for positioning with known displacements, and displacement constraints. The known displacement constraints include zero-value constraints (fixed constraints) or measured displacements (elastic constraints). For fixed boundary conditions, zero-value constraints are added to the corresponding nodal displacement variables to achieve elimination and matrix reduction. For elastic boundary conditions, n measured displacements of the structure are obtained through displacement sensors or other means to add displacement constraints to the elastic boundary structure. This mathematical solution framework eliminates the dependency on boundary condition types, allowing fixed and elastic boundaries to be solved within the same mathematical framework without needing to adjust the algorithm architecture for specific constraint types.

[0041] (3) Numerical methods are used to solve the above-mentioned constrained least squares problem. Optional methods include the Lagrange multiplier method, elimination method, and penalty function method.

[0042] For example, the elimination method is used to solve the problem. The complex geometric model of the raft structure in step (1), which is composed of plates and shells, is refined into a constrained linear least squares problem in step (2).

[0043] For example, the elimination method is chosen as the solution method here. Its core objective is to eliminate known degrees of freedom through constraints, transforming the original problem into a simplified system containing only unknown degrees of freedom. In essence, it is to reconstruct the structure of the system equations and eliminate known degrees of freedom.

[0044] For example, the displacement vector to be determined is decomposed into a known part and an unknown part, and the known displacement is substituted into the representation of the displacement vector, that is:

[0045]

[0046] In the formula This represents the displacement vector to be determined. This indicates the portion of the displacement that is known. This indicates the portion of the displacement that is known. This indicates a known displacement constraint.

[0047] For example, substituting the above equation into the original system equation and rearranging, we get:

[0048]

[0049] To eliminate redundancy in the equation, left multiplication Projected onto a subspace with unknown degrees of freedom:

[0050]

[0051] For example, in order to simultaneously retain known displacement information, equality constraints are added. ,Right now:

[0052]

[0053] The modified equations for the fully free system are:

[0054]

[0055] In the formula This is the corrected stiffness matrix. The corrected load vector is as follows:

[0056]

[0057]

[0058] For example, the displacement can be obtained by inverting the equations of a fully free system. :

[0059]

[0060] Therefore, the improved inverse finite element method can realize full-field displacement reconstruction under arbitrary boundary conditions, breaking through the dependence of existing methods on zero boundary conditions.

[0061] like Figure 1 As shown, a displacement reconstruction method for ship propulsion systems applicable to arbitrary boundary conditions includes the following steps:

[0062] (1) A finite element model of the floating raft structure was performed, nodes were numbered, and strain and displacement measurement points on the raft surface were determined. The weighted least squares error functional between theoretical strain and measured strain was derived using the conventional inverse finite element method, and the following model was obtained after minimizing it. Measurement points with the same distribution as the finite element model were selected on the floating raft for experiments, and displacement and strain sensors were arranged to measure the displacement and strain data.

[0063] (2) Construct a mathematical solution model for reconstructing the displacement field under arbitrary boundary conditions by combining displacement measurement data and strain measurement data.

[0064] (3) To solve the above problem, using measured data as constraints, the system equations are reconstructed using the elimination method to eliminate known degrees of freedom. The displacement vector to be determined is decomposed into known and unknown parts, and the known displacements are substituted into the representation of the displacement vector. The expression is then substituted into the original system equations and rearranged to obtain the modified full-degree-of-freedom system equations. The full-field reconstructed displacement is obtained by combining the corrected stiffness matrix and the load vector.

[0065] To verify the effectiveness of this invention in reconstructing the full-field displacement, a simulation experiment is conducted using a simple thin-plate structure as an example, with different boundary conditions set for verification. Specific details are as follows:

[0066] The thin-plate structure model has its z-axis perpendicular to the xoy plane and faces outwards. The plate structure is made of steel, and the plate length is... board width , plate thickness ,density Poisson's ratio Young's modulus The model uses a Quadratic Hexahedron mesh with a total of 1500 elements and 10928 nodes. The element type is C3D20.

[0067] To comprehensively evaluate the ability of the improved inverse finite element method to handle different boundary constraints, the following two types of typical boundary condition experimental groups were set up:

[0068] (1) Cantilever boundary: constraint Figure 1 The total degrees of freedom of all nodes on the boundary of path 1;

[0069] (2) Elastic boundary support: Elastic support points are arranged at specific locations on the plate. Four elastic support points are set (elastic support 1 to elastic support 4).

[0070] Under cantilever boundary conditions, the structure's self-weight has already caused significant static deformation at the free end (path 2 boundary node). To effectively isolate the boundary treatment effect and focus on method comparison, gravity loads are ignored in the cantilever boundary. For the elastically supported boundary, the effect of gravity loads is retained. The specific external load settings for each working condition are as follows:

[0071] (1) Cantilever boundary: Apply a vertical concentrated load at node P3.

[0072] (2) Elastic support boundary: A vertical concentrated load is applied at node P1.

[0073] The displacement reconstruction results were verified by comparison with the high-precision ANSYS finite element simulation reference solution. First, a finite element model was constructed using ANSYS, the thin plate structure was discretized, and inverse elements were generated. When different z-axis loads were applied at points P1 to P4, the displacement reconstruction results were then compared with the high-precision ANSYS finite element simulation reference solution. Figure 2 The measured x-direction strain data of S1~S10 and the measured z-direction displacement data of N1~N18 were used as actual constraints under arbitrary boundary conditions. These data were substituted into the original system equations of the inverse finite element method, and the system equations were reconstructed using the elimination method. Known degrees of freedom were eliminated, resulting in the corrected fully free system equations. The reconstructed displacements were then solved by inverting the corrected stiffness matrix and multiplying it by the corrected load vector. Subsequently, the reconstructed displacements were compared with the vertical displacements obtained from finite element analysis and the conventional inverse finite element method.

[0074] Calculation process description:

[0075] The first step is to construct a finite element model of the raft structure and perform inverse element division. Experiments are then conducted, with displacement sensors and strain sensors arranged at various measuring points of the thin plate structure under arbitrary boundary conditions. Loads are applied at designated nodes, and the displacement and strain of the thin plate are measured to obtain displacement and strain measurements.

[0076] The second step involves using the displacement and strain data obtained in the first step as known constraints in the system equations to construct a constrained linear least squares problem-solving model.

[0077] The third step involves obtaining the corrected stiffness matrix and load vector using the elimination method. The full-field reconstructed displacement values ​​under arbitrary boundary conditions are obtained by inverting the corrected stiffness matrix and right-multiplying it by the corrected load vector. The comparison results between the reconstructed displacement and the reference displacement of the cantilever structure are shown below. Figure 3 As shown, the comparison results between the reconstructed displacement and the reference displacement of the elastic support structure are as follows: Figure 4 As shown.

[0078] The above description provides examples of the preferred embodiments of the present invention. Parts not detailed herein are common knowledge to those skilled in the art. The scope of protection of the present invention is determined by the claims. Any equivalent modifications based on the technical teachings of the present invention are also within the scope of protection of the present invention.

Claims

1. A displacement reconstruction method based on an improved inverse finite element method applicable to raft structures with arbitrary boundaries, characterized in that, Includes the following steps: Step 1: Under the condition of elastic boundary, determine the specifications, quantity and position of the elastic elements on the solid floating raft, arrange strain measuring points and displacement measuring points on the upper and lower surfaces of the solid floating raft, apply loads at designated nodes of the solid floating raft, measure the strain and displacement of the solid floating raft, and obtain the strain measurement data and displacement measurement data of the solid floating raft. Under fixed boundary conditions, strain measurement points are arranged on the upper and lower surfaces of the solid floating raft, and loads are applied at designated nodes of the solid floating raft to measure the strain of the solid floating raft and obtain strain measurement data of the solid floating raft. Step 2: Calculate the global stiffness matrix based on the strain measurement data of the solid floating raft. and class load vector ; Step 3: Combining measured displacement and strain data, the constrained linear least squares problem under arbitrary boundary conditions is obtained: ； In the formula This represents the displacement vector to be determined. express The estimated value, Given the location Boolean matrix of the measured displacement data, under elastic boundary conditions, For actual displacement measurement data, under fixed boundary conditions, =0; Step 4: Solve the above constrained least squares problem using numerical methods to obtain the optimal solution U. This is the full-field reconstruction displacement.

2. The displacement reconstruction method for raft structures with arbitrary boundaries based on the improved inverse finite element method according to claim 1, characterized in that, Step 1 specifically includes the following steps: A grid is divided on the upper and lower surfaces of the solid floating raft, and strain and displacement measuring points are arranged at the nodes of the grid. The number of strain measuring points on the upper and lower surfaces of the solid floating raft is the same and they correspond one-to-one.

3. The displacement reconstruction method for raft structures with arbitrary boundaries based on the improved inverse finite element method according to claim 1, characterized in that, In step 2, the global stiffness matrix and class load vector It is calculated using the following formula: ； In the formula, Indicates membrane strain, Indicates the curvature of the curve. , , Let be the displacement-strain transformation matrices for the corresponding components. , , These are the error weighting coefficients for the membrane strain, bending curvature, and transverse shear strain components, respectively. , , , where n is the total number of strain measurement points on the upper or lower surface of the floating raft, and the measured strains at the i-th strain measurement point on the upper or lower surface of the solid floating raft are respectively and Then the membrane strain at the i-th mid-surface and curvature The calculation formula is: 。

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