Composite material cracking path calculation method based on matrix-free pixel finite element

Abstract

The invention provides a composite material cracking path calculation method based on a matrix-free pixel finite element, pixels of a composite material image are used as fine grids of the finite element, and different pixel gray values represent different material components. And coarse graining the fine grids to form coarse grids, and embedding the boundaries of the material components into the coarse grids to form embedded boundary units. A stiffness matrix of a coarse grid unit is calculated through Gaussian integral, the overall stiffness does not need to be assembled, and coarse grid node displacement is solved through a conditional conjugate gradient algorithm. And on the basis of the coarse grid node displacement, fine grid node displacement is obtained through an interpolation method, and then fine grid unit stress is obtained. Taking a maximum tensile stress criterion as a damage criterion of the unit, taking a unit region as a crack after the unit is damaged, adding a corresponding substance unit at an image boundary, and keeping substance conservation. According to the calculation method, little computer memory is occupied, and the requirement of cracking calculation for hardware resources is greatly reduced.

Description

Technical Field

[0001] The invention relates to the field of composite materials, and in particular to a method for calculating a cracking path of a composite material based on matrix-free pixel finite elements. Background Art

[0002] Composite materials will crack under load. The calculation methods of composite material cracking mainly include finite element model, cohesive force model and extended finite element. These calculation methods usually use composite material images to construct cracking calculation models, assemble the overall stiffness matrix through unit stiffness matrix, and then solve the displacement of unit nodes to obtain the stress of units. However, composite material images are composed of a large number of pixels or voxels, so that the composite material cracking calculation model established by composite material images can reach millions of degrees of freedom, resulting in a huge scale of finite element overall stiffness matrix, causing the memory space required for the calculation of composite material cracking model to approach or even exceed the limit of personal computers. Therefore, the calculation of cracking model often needs to rely on powerful hardware computing resources, causing hardware computing resources to become the bottleneck of composite material cracking model calculation. Summary of the invention

[0003] In view of the above-mentioned defects of the prior art, the technical problem to be solved by the present invention is that the existing cracking model calculation needs to rely on powerful hardware computing resources, which makes hardware computing resources become the bottleneck of the calculation of the composite material cracking model. The present invention provides a composite material cracking path calculation method based on matrix-free pixel finite element, which greatly reduces the computing memory space requirement, meets the computing requirements and improves the computing accuracy.

[0004] To achieve the above object, the present invention provides a method for calculating the crack path of a composite material based on matrix-free pixel finite element, comprising the following steps:

[0005] Step 1: Take composite material images with a smartphone and perform preprocessing;

[0006] Step 2: Establish a matrix-free pixel finite element fine mesh model;

[0007] Step 3: doubling the size of the matrix-free pixel finite element fine grid to obtain a matrix-free pixel finite element coarse grid model, and embedding the boundary of the composite material into the unit to form an embedded boundary unit;

[0008] Step 4: Calculate the geometric matrix, Jacobian matrix, stiffness matrix of each material attribute fine mesh unit, and stress function matrix of each material attribute fine mesh unit of the matrix-free pixel finite element fine mesh unit according to the unit node coordinates of the matrix-free pixel finite element fine mesh;

[0009] Step 5: Calculate the number of the fine grid cell included in the coarse grid cell according to the dependency relationship between the coarse grid cell number and the fine grid cell number, identify the material properties of the fine grid cell based on the fine grid cell number, and calculate the stiffness matrix of the coarse grid cell using the average method based on the stiffness matrix calculated based on the material properties of the fine grid cell;

[0010] Step 6: Use the unit stiffness matrix of the coarse grid and the diagonal matrix of its stiffness matrix to assemble the pre-processing matrix of the pre-processing conjugate gradient method;

[0011] Step 7: Apply displacement loads to the left and right boundaries of the composite material image, and calculate the forces acting on the left and right boundaries of the image through the stress function matrix, displacement load rate, number of cycles and time step;

[0012] Step 8: Use the pre-conditioned conjugate gradient method to calculate the displacement of the coarse grid nodes;

[0013] Step 9: Calculate the displacement of the fine grid nodes by interpolation based on the displacement of the coarse grid nodes;

[0014] Step 10: Calculate the stress of the fine grid by using the node displacement and stress function matrix of the fine grid;

[0015] Step 11: When the stress of the fine grid unit exceeds the limit stress of the composite material component, the fine grid unit is destroyed. After the unit is destroyed, the unit area is regarded as a crack, and the corresponding material unit is added to the air layer of the image to maintain the material conservation of the calculation model;

[0016] Repeat steps 7 to 11 until connected cracks are observed in the composite material image, then stop the calculation process and obtain the cracking path of the composite material.

[0017] Furthermore, step one, obtaining a composite material image and performing preprocessing, specifically includes obtaining a composite material image, segmenting the obtained composite material image, obtaining the distribution of each component of the composite material, and assigning a specific gray value to each component.

[0018] Furthermore, an air layer is added at the boundary of the composite material image and assigned a gray value of 175.

[0019] Furthermore, in step two, a matrix-free pixel finite element fine mesh model is established, wherein the matrix-free pixel finite element fine mesh model includes the number of pixels in the x-direction, the number of pixels in the y-direction, the pixel grayscale value, the material type, the material properties, the number of fine mesh nodes, the number of fine mesh units, the degree of freedom value, and the unit type.

[0020] Furthermore, in step three, the matrix-free pixel finite element coarse grid model includes the number of coarse grids in the x-direction, the number of coarse grids in the y-direction, the iteration allowable error, the number of iterations, the number of coarse grid nodes, and the number of coarse grid units.

[0021] Further, in step 4, according to the unit node coordinates of the matrix-free pixel finite element fine grid, using

[0022] Formula 1 calculates the Jacobian matrix J of the matrix-free pixel finite element fine grid unit

[0023]

[0024] Among them, x and y are the coordinates of the physical coordinate system; ξ and η are the coordinates of the projected coordinate system.

[0025] Furthermore, in step 4, according to the unit node coordinates of the matrix-free pixel finite element fine grid, the geometric matrix B of the matrix-free pixel finite element fine grid unit is calculated using Formula 2,

[0026]

[0027] Among them, ξ and η are the coordinates of the projected coordinate system; J -1 is the inverse of the Jacobian matrix; N * (ξ,η) is the shape function matrix.

[0028] Furthermore, in step 4, according to the unit node coordinates of the matrix-free pixel finite element fine grid, the fine grid unit stiffness matrix K of each material attribute is e

[0029]

[0030] Where t is the thickness of the plane problem, t = 1

[0031] Furthermore, in step 4, according to the unit node coordinates of the matrix-free pixel finite element fine grid, the stress function matrix S of each material attribute fine grid unit is calculated using formula (4):

[0032] S=CB (4)

[0033] Where C is the elastic coefficient matrix.

[0034] Furthermore, in step 5, according to the dependency relationship between the coarse grid unit number and the fine grid unit number, the number of the fine grid unit included in the coarse grid unit is calculated, the material properties of the fine grid unit are identified based on the number of the fine grid unit, and the stiffness matrix of the coarse grid unit is calculated according to the material properties of the fine grid unit. The average method is used to calculate the stiffness matrix of the coarse grid unit, which specifically includes the stiffness matrix K calculated according to the material properties of the fine grid unit.e , use formula (5) to calculate the stiffness matrix of the coarse grid unit.

[0035]

[0036] Where: is the stiffness matrix of the fine mesh element; J i is a constant Jacobian matrix, and the calculation formula is:

[0037]

[0038] Where: h1 and h2 are the dimensions of the pixel in the x and y directions.

[0039] Technical Effects

[0040] The present invention provides a method for calculating the crack path of composite materials based on matrix-free pixel finite elements. It does not require the assembly of an overall stiffness matrix, which reduces the demand for computer memory space during the finite element calculation process, effectively solves the problem of high degree of freedom in calculation, and realizes a low-memory cracking calculation method based on composite material images. Embedded boundary units are introduced during the calculation process to improve the calculation accuracy of the matrix-free pixel finite element method for composite material cracking.

[0041] The concept, specific structure and technical effects of the present invention will be further described below in conjunction with the accompanying drawings to fully understand the purpose, characteristics and effects of the present invention. BRIEF DESCRIPTION OF THE DRAWINGS

[0042] Figure 1 It is a flow chart of a method for calculating cracking path of composite materials based on matrix-free pixel finite element according to a preferred embodiment of the present invention;

[0043] Figure 2 The present invention is a composite material crack path diagram of a composite material crack path calculation method based on matrix-free pixel finite element according to a preferred embodiment of the present invention. DETAILED DESCRIPTION

[0044] In order to make the technical problems, technical solutions and beneficial effects to be solved by the present invention more clearly understood, the present invention is further described in detail below in conjunction with the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are only used to explain the present invention and are not intended to limit the present invention.

[0045] In the following description, specific details such as specific internal procedures and techniques are provided for the purpose of illustration rather than limitation, so as to provide a thorough understanding of the embodiments of the present invention. However, it should be clear to those skilled in the art that the present invention may be implemented in other embodiments without these specific details. In other cases, detailed descriptions of well-known systems, devices, circuits, and methods are omitted to prevent unnecessary details from obstructing the description of the present invention.

[0046] like Figure 1 As shown, the present invention provides a method for calculating the cracking path of a composite material based on matrix-free pixel finite element, comprising the following steps:

[0047] Step 1: Take composite material images with a smartphone and perform preprocessing; specifically:

[0048] The coarse aggregate and void distribution of the composite image taken by the smartphone was obtained by using the im2bw function in MATLAB. The coarse aggregate and void distribution was subtracted from the composite image using the imsubtract function in MATALB to obtain the fine aggregate asphalt concrete distribution in the composite image. MATLAB was used to assign grayscale values ​​of 255, 125, and 0 to the three components of the composite. MATLAB was used to add an air layer to the boundary of the composite image, and the air layer was assigned a grayscale value of 175. The purpose of adding the air layer is to store the material corresponding to the cracking unit and maintain the conservation of the model material.

[0049] Step 2: Establish a matrix-free pixel finite element fine mesh model; specifically, it includes creating a data structure of a matrix-free pixel finite element fine mesh, which mainly includes the sum of the number of pixels in the x-direction image and the air layer is 640, the sum of the number of pixels in the y-direction image and the air layer is 640, the pixel grayscale values ​​are 255, 175, 125 and 0, there are 3 types of materials, the number of fine mesh nodes is 410881, the number of fine mesh units is 409600, the degree of freedom value is 821762, and the material properties are shown in Table 1.

[0050] Table 1 Material parameters

[0051]

[0052] Step 3: Double the size of the matrix-free pixel finite element fine grid to obtain a matrix-free pixel finite element coarse grid model, and embed the boundary of the composite material into the unit to form an embedded boundary unit; the specific steps are as follows: Create a data structure of the matrix-free pixel finite element coarse grid, which mainly includes 320 coarse grids in the x direction, 320 coarse grids in the y direction, 0.000001 iteration allowable error, 1000 iterations, 103041 coarse grid nodes, and 102400 coarse grid units. Set the size of the coarse grid in the x and y directions to twice the size of the fine grid in the x and y directions, so as to properly accommodate the boundary of the fine grid unit in the coarse grid to form an embedded boundary unit.

[0053] Step 4: Project the coordinates of the fine mesh unit nodes to the reference coordinate system, use formula (1) to calculate the Jacobian matrix J of the fine mesh unit, use formula (2) to calculate the fine mesh geometry matrix B, and use formula (3) to calculate the fine mesh unit stiffness matrix K of each material property e , use formula (4) to calculate the stress function matrix S of the fine grid unit of each material attribute.

[0054]

[0055]

[0056]

[0057] S=CB (4)

[0058] Where: x and y are the coordinates of the physical coordinate system; ξ and η are the coordinates of the projected coordinate system; J -1 is the inverse of the Jacobian matrix; N * (ξ,η) is the shape function matrix; C is the elastic coefficient matrix; t is the thickness of the plane problem, t=1.

[0059] Step 5: According to the dependency relationship between the coarse grid unit number and the fine grid unit number, calculate the number of the fine grid unit contained in the coarse grid unit, and identify the material properties of the fine grid unit based on the number of the fine grid unit. The stiffness matrix K calculated based on the material properties of the fine grid unit e , use formula (5) to calculate the stiffness matrix of the coarse grid unit.

[0060]

[0061] Where: is the stiffness matrix of the fine mesh element; J i is a constant Jacobian matrix, and the calculation formula is:

[0062]

[0063] Where: h1 and h2 are the dimensions of the pixel in the x and y directions.

[0064] Step 6: Use the unit stiffness matrix of the coarse grid and the diagonal matrix of its stiffness matrix to assemble according to the positions corresponding to the node displacements, so as to construct the preprocessing matrix M of the preprocessing conjugate gradient method.

[0065] M=diag{K 11 ,K 22 ,…,K nn} (7)

[0066] Step 7: Apply displacement loads to the left and right boundaries of the composite material image, and calculate the forces acting on the left and right boundaries of the image through the stress function matrix, displacement load rate, number of cycles, and time step. Specifically, apply a displacement load of 50 mm / min to the left and right boundaries of the image, and calculate the forces acting on the left and right boundaries of the image through the unit matrix of the coarse grid, the displacement load rate of 50 mm / min, the number of cycles n, and the time step of 0.00001 s.

[0067] Step 8: Use the preconditioned conjugate gradient method algorithm 1 to iteratively calculate the displacement of the coarse grid nodes. When the iteration error is less than 0.000001, the displacement of the four nodes of the coarse grid is obtained.

[0068] Algorithm 1 Preconditioned conjugate gradient method

[0069] 1: Select the initial displacement vector q e

[0070] 2: Residual vector r = fK c q e

[0071] 3: Search direction d = M -1 r

[0072] 4: Scalar parameter δ1 = r T d

[0073] 5: Scalar parameter δ0 = δ1

[0074] 6:while(δ1 / δ0>ε 2 )do

[0075] 7: Auxiliary vector h = Kd

[0076] 8: Step size α = δ1 / d T h

[0077] 9: Solved displacement vector x = x + αd

[0078] 10: Residual vector r = r - αh

[0079] 11: Residual s of preprocessing space = M -1 r

[0080] 12: Scalar parameter δ2 = δ1

[0081] 13: Scalar parameter δ1 = r T s

[0082] 14: Calculate the factor β of the conjugate search direction = δ1 / δ2

[0083] 15: Search direction d = s + βd

[0084] 16: end while

[0085] In Algorithm 1: f is the load; ε is the iteration error.

[0086] Step 9: The nodes of the coarse grid overlap with the nodes of the fine grid, and the displacement of the coarse grid node is assigned to the corresponding fine grid node; the midpoint of the coarse grid edge is the fine grid node, and the average value of the sum of the displacements of the two nodes on the coarse grid edge is taken as its displacement value; the center point of the coarse grid is the fine grid node, and one quarter of the sum of the displacements of the four nodes of the coarse grid is taken as its displacement value, thereby obtaining the displacements of the nine fine grid nodes contained in the coarse grid.

[0087] Step 10: Apply the fine mesh node displacements q e And the stress function matrix S, calculate the stress of the fine grid.

[0088] σ=Sq e (8)

[0089] Step 11: When the stress of the fine grid unit exceeds the limit stress of the composite material component, the fine grid unit is destroyed. After the fine grid unit is destroyed, the fine grid area is regarded as a crack, and the corresponding fine grid material component is added to the air layer of the image to maintain the conservation of matter in the calculation model.

[0090] Step 12: Repeat steps 7 to 11 until connected cracks are observed in the composite material image, then stop the calculation process and obtain the cracking of the composite material, such as Figure 2 shown.

[0091] The preferred specific embodiments of the present invention are described in detail above. It should be understood that a person skilled in the art can make many modifications and changes based on the concept of the present invention without creative work. Therefore, any technical solution that can be obtained by a person skilled in the art through logical analysis, reasoning or limited experiments based on the concept of the present invention on the basis of the prior art should be within the scope of protection determined by the claims.

Claims

1. A method for calculating cracking paths of composite materials based on matrix-free pixel finite element, characterized in that: The following steps are involved: Step 1: Obtain composite material images and perform preprocessing; Step 2: Establish a matrix-free pixel finite element fine mesh model; Step 3: doubling the size of the matrix-free pixel finite element fine grid to obtain a matrix-free pixel finite element coarse grid model, and embedding the boundary of the composite material into the unit to form an embedded boundary unit; Step 4: Calculate the geometric matrix, Jacobian matrix, stiffness matrix of each material attribute fine mesh unit, and stress function matrix of each material attribute fine mesh unit of the matrix-free pixel finite element fine mesh unit according to the unit node coordinates of the matrix-free pixel finite element fine mesh; Step 5: Calculate the number of the fine grid cell included in the coarse grid cell according to the dependency relationship between the coarse grid cell number and the fine grid cell number, identify the material properties of the fine grid cell based on the fine grid cell number, and calculate the stiffness matrix of the coarse grid cell using the average method based on the stiffness matrix calculated based on the material properties of the fine grid cell; Step 6: Use the unit stiffness matrix of the coarse grid and the diagonal matrix of its stiffness matrix to assemble the pre-processing matrix of the pre-processing conjugate gradient method; Step 7: Apply displacement loads to the left and right boundaries of the composite material image, and calculate the forces acting on the left and right boundaries of the image through the stress function matrix, displacement load rate, number of cycles and time step; Step 8: Use the pre-conditioned conjugate gradient method to calculate the displacement of the coarse grid nodes; Step 9: Calculate the displacement of the fine grid nodes by interpolation based on the displacement of the coarse grid nodes; Step 10: Calculate the stress of the fine grid by using the node displacement and stress function matrix of the fine grid; Step 11: When the stress of the fine grid unit exceeds the limit stress of the composite material component, the fine grid unit is destroyed. After the unit is destroyed, the unit area is regarded as a crack, and the corresponding material unit is added to the air layer of the image to maintain the material conservation of the calculation model; Repeat steps 7 to 11 until connected cracks are observed in the composite material image, then stop the calculation process and obtain the cracking path of the composite material.

2. A method for calculating cracking paths of composite materials based on matrix-free pixel finite element as claimed in claim 1, characterized in that: Step 1: Acquire a composite material image and perform preprocessing, specifically including acquiring a composite material image, segmenting the acquired composite material image, obtaining the distribution of each component of the composite material, and assigning a specific grayscale value to each component.

3. A method for calculating cracking paths of composite materials based on matrix-free pixel finite element as claimed in claim 2, characterized in that: Add an air layer at the boundary of the composite image and assign a grayscale value of 175.

4. A method for calculating cracking paths of composite materials based on matrix-free pixel finite element as claimed in claim 1, characterized in that: Step 2: Establish a matrix-free pixel finite element fine mesh model, wherein the matrix-free pixel finite element fine mesh model includes the number of pixels in the x-direction, the number of pixels in the y-direction, the pixel grayscale value, the material type, the material properties, the number of fine mesh nodes, the number of fine mesh units, the degree of freedom value, and the unit type.

5. The method for calculating cracking path of composite materials based on matrix-free pixel finite element as claimed in claim 1, characterized in that: Step 3: The matrix-free pixel finite element coarse grid model includes the number of coarse grids in the x-direction, the number of coarse grids in the y-direction, the iteration allowable error, the number of iterations, the number of coarse grid nodes, and the number of coarse grid units.

6. A method for calculating cracking paths of composite materials based on matrix-free pixel finite element as claimed in claim 1, characterized in that: In step 4, according to the unit node coordinates of the matrix-free pixel finite element fine grid, the Jacobian matrix J of the matrix-free pixel finite element fine grid unit is calculated using Formula 1; Among them, x and y are the coordinates of the physical coordinate system; ξ and η are the coordinates of the projected coordinate system.

7. A method for calculating cracking paths of composite materials based on matrix-free pixel finite element as claimed in claim 6, characterized in that: In step 4, according to the unit node coordinates of the matrix-free pixel finite element fine grid, the geometric matrix B of the matrix-free pixel finite element fine grid unit is calculated using Formula 2: Where ξ and η are the coordinates of the projected coordinate system; J -1 is the inverse of the Jacobian matrix; N * (ξ,η) is the shape function matrix.

8. A method for calculating cracking paths of composite materials based on matrix-free pixel finite element as claimed in claim 7, characterized in that: In step 4, according to the unit node coordinates of the matrix-free pixel finite element fine grid, the fine grid unit stiffness matrix K of each material attribute is e Where t is the thickness of the plane problem, t=1.

9. A method for calculating cracking paths of composite materials based on matrix-free pixel finite element as claimed in claim 8, characterized in that: In step 4, according to the unit node coordinates of the matrix-free pixel finite element fine grid, the stress function matrix S of each material attribute fine grid unit is calculated using formula (4): S=CB (4) Where C is the elastic coefficient matrix.

10. The method for calculating cracking path of composite materials based on matrix-free pixel finite element as claimed in claim 8, characterized in that: In step 5, according to the dependency relationship between the coarse grid unit number and the fine grid unit number, the number of the fine grid unit contained in the coarse grid unit is calculated, the material properties of the fine grid unit are identified based on the number of the fine grid unit, and the stiffness matrix of the coarse grid unit is calculated based on the material properties of the fine grid unit. The average method is used to calculate the stiffness matrix of the coarse grid unit, which specifically includes the stiffness matrix K calculated based on the material properties of the fine grid unit e , use formula (5) to calculate the stiffness matrix of the coarse grid unit, Where: is the stiffness matrix of the fine mesh element; J i is a constant Jacobian matrix, and the calculation formula is: Where: h1 and h2 are the dimensions of the pixel in the x and y directions.

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