A composite cracking path calculation method based on matrix-free pixel finite element

Abstract

The application provides a composite cracking path calculation method based on a matrix-free pixel finite element, in which pixels of a composite image are fine meshes of finite elements, and different pixel gray values represent different material components. After coarse-grained of the fine meshes, coarse meshes are formed, the boundaries of the material components are embedded into the interior of the coarse meshes to form embedded boundary elements. The stiffness matrix of the coarse mesh elements is calculated through Gaussian integration, and the overall stiffness is not assembled. The conditional conjugate gradient algorithm is used to solve the node displacement of the coarse mesh. Based on the node displacement of the coarse mesh, the node displacement of the fine mesh is obtained through interpolation, and then the stress of the fine mesh element is obtained. The maximum tensile stress criterion is used as the element failure criterion. After the element is damaged, the element area is regarded as a crack, and corresponding material elements are added to the image boundary to maintain the conservation of mass. The calculation method of the application occupies very little computer memory and greatly reduces the demand for hardware resources for cracking calculation.

Description

TECHNICAL FIELD

[0001] The present application relates to the field of composite materials, and particularly relates to a composite material cracking path calculation method based on a matrix-free pixel finite element. BACKGROUND

[0002] Under the action of load, the composite material will produce cracking phenomenon, and the calculation method of the composite material cracking mainly includes a finite element model, a cohesive force model and an extended finite element, etc. These calculation methods usually construct a cracking calculation model by using a composite material image, assemble an overall stiffness matrix by using an element stiffness matrix, and then solve the displacement of an element node to obtain the stress of the element. However, the composite material image is composed of a large number of pixels or voxels, so that the composite material cracking calculation model established by the composite material image can reach several million degrees of freedom, resulting in that the overall stiffness matrix of the finite element is large in size, and the required memory space for the calculation of the composite material cracking model is close to or even exceeds the limit of a personal computer. Therefore, the calculation of the cracking model often needs to rely on powerful hardware computing resources, and the hardware computing resources become the bottleneck of the calculation of the composite material cracking model. SUMMARY

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

[0004] To achieve the above-mentioned purpose, the present application provides a composite material cracking path calculation method based on a matrix-free pixel finite element, comprising the following steps:

[0005] Step one, capturing a composite material image by using a smart phone and performing preprocessing;

[0006] Step two, establishing a matrix-free pixel finite element fine grid model;

[0007] Step three, enlarging the size of the matrix-free pixel finite element fine grid by one time to obtain a matrix-free pixel finite element coarse grid model, and embedding the boundary of the composite material into the element interior to form an embedded boundary element;

[0008] Step four, calculating the geometric matrix, the Jacobian matrix, the fine grid element stiffness matrix of each material property and the fine grid element stress function matrix of each material property of the matrix-free pixel finite element fine grid element according to the element node coordinates of the matrix-free pixel finite element fine grid;

[0009] Step five, according to the dependence of the coarse grid cell number and the fine grid cell number, calculate the number of the fine grid cell contained in the coarse grid cell, identify the material properties of the fine grid cell based on the number of the fine grid cell, calculate the stiffness matrix of the fine grid cell according to the material properties of the fine grid cell, and calculate the stiffness matrix of the coarse grid cell by using the average method;

[0010] Step six, using the element stiffness matrix of the coarse grid, using the diagonal matrix of its stiffness matrix to assemble the preconditioning matrix of the preconditioned conjugate gradient method;

[0011] Step seven, apply displacement load on the left and right boundaries of the composite image, calculate the force acting on the left and right boundaries of the image by the stress function matrix, displacement load rate, cycle number and time step;

[0012] Step eight, using the preconditioned conjugate gradient method, calculate the displacement of the coarse grid node;

[0013] Step nine, according to the displacement of the coarse grid node, calculate the displacement of the fine grid node by interpolation method;

[0014] Step ten, using the node displacement and stress function matrix of the fine grid, calculate the stress of the fine grid;

[0015] Step eleven, when the stress of the fine grid element exceeds the ultimate stress of the composite component, the fine grid element is damaged, and the element area is regarded as a crack after the element is damaged, corresponding material elements are added in the air layer of the image, and the conservation of mass of the calculation model is maintained;

[0016] Repeat steps seven to eleven until a connected crack is observed in the composite image, then stop the calculation process and obtain the cracking path of the composite material.

[0017] Further, step one, obtaining a composite material image and pre-processing, specifically including obtaining a composite material image, segmenting the obtained composite material image to obtain the distribution of each component of the composite material, and assigning a specific gray value to each component.

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

[0019] Further, step two, establishing a matrix-free pixel finite element fine grid model, wherein the matrix-free pixel finite element fine grid model includes the number of x-direction pixels, the number of y-direction pixels, pixel gray value, material type, material property, fine grid node number, fine grid element number, degree of freedom value, and element type.

[0020] Further, in step three, the matrix-free pixel finite element coarse grid model comprises the number of coarse grid nodes in the x direction, the number of coarse grid nodes in the y direction, the iteration allowed error, the iteration number, the number of coarse grid nodes, and the number of coarse grid elements.

[0021] Further, in step four, according to the element node coordinates of the matrix-free pixel finite element fine grid, the geometric matrix B of the matrix-free pixel finite element fine grid element is calculated by using formula 2.

[0022] Formula 1 is used to calculate the Jacobian matrix J of the matrix-free pixel finite element fine grid element.

[0023]

[0024] Wherein, x and y are the coordinates of the physical coordinate system; ξ and η are the coordinates of the projection coordinate system.

[0025] Further, in step four, according to the element node coordinates of the matrix-free pixel finite element fine grid, the geometric matrix B of the matrix-free pixel finite element fine grid element is calculated by using formula 2.

[0026]

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

[0028] Further, in step four, according to the element node coordinates of the matrix-free pixel finite element fine grid, the stiffness matrix K of each material attribute fine grid element is calculated. e

[0029]

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

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

[0032] S = CB (4)

[0033] Wherein, C is the elastic coefficient matrix.

[0034] Further, in step five, according to the dependence of the coarse grid element number and the fine grid element number, the number of fine grid elements contained in the coarse grid element is calculated, the material properties of the fine grid elements are identified based on the number of fine grid elements, the stiffness matrix of the fine grid elements is calculated according to the material properties of the fine grid elements, and the average method is used to calculate the stiffness matrix of the coarse grid element.e The stiffness matrix of the coarse grid unit is calculated by using formula (5).

[0035]

[0036] In the formula, h1 and h2 are the sizes of the pixels in the x and y directions. is the stiffness matrix of the fine grid unit; J i is a constant Jacobian matrix, and the calculation formula is:

[0037]

[0038] In the formula, h1 and h2 are the sizes of the pixels in the x and y directions.

[0039] Technical effects

[0040] The composite material cracking path calculation method based on the matrix-free pixel finite element provided by the application does not need to assemble an overall stiffness matrix, reduces the demand for computer memory space in the finite element calculation process, effectively solves the problem of high degree of freedom in calculation, realizes a low-memory-occupancy cracking calculation method based on a composite material image, and introduces an embedded boundary unit in the calculation process, thereby improving 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 application will be further described below with reference to the drawings, so as to fully understand the purposes, features and effects of the application. BRIEF DESCRIPTION OF DRAWINGS

[0042] Figure 1 is a flowchart of a composite material cracking path calculation method based on a matrix-free pixel finite element of a preferred embodiment of the application;

[0043] Figure 2 is a composite material cracking path diagram of a composite material cracking path calculation method based on a matrix-free pixel finite element of a preferred embodiment of the application. DETAILED DESCRIPTION

[0044] In order to make the technical problems, technical solutions and beneficial effects of the application clearer, the application will be further described in detail below with reference to the drawings and embodiments. It should be understood that the specific embodiments described herein are only used to explain the application and do not limit the application.

[0045] In the following description, for purposes of explanation and not limitation, specific details are set forth such as particular procedures, techniques, etc. in order to provide a thorough understanding of the embodiments of the present application. However, it will be apparent to those skilled in the art that the present application can be practiced in other embodiments that depart from these specific details. In other instances, detailed descriptions of well-known methods, devices, and circuits are omitted so as not to obscure the description of the present application with unnecessary detail.

[0046] As shown in Figure 1 The present application provides a composite cracking path calculation method based on matrix-free pixel finite element, comprising the following steps:

[0047] Step one: taking a composite image by a smart phone and pre-processing; specifically including:

[0048] The coarse aggregate and void distribution of the composite image taken by the smart phone is obtained by the im2bw function in MATLAB, the fine aggregate asphalt concrete distribution in the composite image is obtained by subtracting the coarse aggregate and void distribution from the composite image by the imsubtract function in MATLAB. The three components of the composite material are given gray values of 255, 125 and 0 by MATLAB. An air layer is added to the boundary of the composite image by MATLAB, and the air layer is given a gray value of 175. The function of adding an air layer is to store the corresponding substances of the cracking unit and maintain the conservation of model substances.

[0049] Step two: establishing a matrix-free pixel finite element fine grid model; specifically including creating a data structure body of the matrix-free pixel finite element fine grid, which mainly includes the sum of the number of x-direction image and air layer pixels is 640, the sum of the number of y-direction image and air layer pixels is 640, the pixel gray value is 255, 175, 125 and 0, the material type is 3, the fine grid node number is 410881, the fine grid element number is 409600, the degree of freedom value is 821762, and the material attribute is as shown in Table 1.

[0050] Table 1 Material parameters

[0051]

[0052] Step three: the size of the matrixless pixel finite element fine mesh is doubled to obtain a matrixless pixel finite element coarse mesh model, and the boundary of the composite material is embedded into the unit to form an embedded boundary unit; the specific steps are as follows: a data structure body of the matrixless pixel finite element coarse mesh is created, which mainly includes that the number of coarse meshes in the x direction is 320, the number of coarse meshes in the y direction is 320, the iteration allowed error is 0.000001, the iteration number is 1000 times, the number of coarse mesh nodes is 103041, and the number of coarse mesh units is 102400. The x direction and y direction size of the coarse mesh is set to be twice the size of the x direction and y direction size of the fine mesh, so as to properly accommodate the boundary of the fine mesh unit in the coarse mesh to form an embedded boundary unit.

[0053] Step four: the coordinates of the fine mesh unit nodes are projected to the reference coordinate system, the Jacobian matrix J of the fine mesh unit is calculated by formula (1), the fine mesh geometric matrix B is calculated by formula (2), the stiffness matrix K of the fine mesh unit of each material property is calculated by formula (3), and the stress function matrix S of the fine mesh unit of each material property is calculated by formula (4). e

[0054]

[0055]

[0056]

[0057] S=CB (4)

[0058] In the formula: 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 five: according to the dependence relationship between the coarse mesh unit number and the fine mesh unit number, the number of fine mesh units contained in the coarse mesh unit is calculated, and the material properties of the fine mesh unit are identified based on the number of fine mesh units. The stiffness matrix K e calculated according to the material properties of the fine mesh unit is used to calculate the stiffness matrix of the coarse mesh unit by formula (5).

[0060]

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

[0062] ​

[0063] where h1 and h2 are the size of the pixels in x and y direction.

[0064] Step six: using the element stiffness matrix of the coarse mesh, the diagonal matrix of its stiffness matrix is used to assemble according to the position corresponding to the node displacement, so as to construct the preconditioning matrix M of the preconditioned conjugate gradient method.

[0065] M = diag{K 11 ,K 22 ,…,K nn} (7)

[0066] Step seven: displacement load is applied on the left and right boundaries of the composite image, and the force acting on the left and right boundaries of the image is calculated by the stress function matrix, displacement load rate, cycle number and time step. Specifically, a displacement load of 50mm / min is applied on the left and right boundaries of the image, and the force acting on the left and right boundaries of the image is calculated by the element matrix of the coarse mesh, displacement load rate 50mm / min, cycle number n and time step 0.00001s.

[0067] Step eight: using the preconditioned conjugate gradient method algorithm 1, the displacement of the coarse mesh nodes is calculated in an iterative manner, and when the iteration error is less than 0.000001, the displacement of the four nodes of the coarse mesh is obtained.

[0068] Algorithm 1: preconditioned conjugate gradient method

[0069] 1: select initial displacement vector q e

[0070] 2: residual vector r = f - K 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: solution displacement vector x = x + αd

[0078] 10: residual vector r = r - αh

[0079] 11: Preconditioned residual s = M -1 r

[0080] 12: Scalar parameter delta2 = delta1

[0081] 13: Scalar parameter delta1 = r T s

[0082] 14: Compute factor beta = delta1 / delta2 for conjugate search direction

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

[0084] 16: end while

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

[0086] Step nine: the nodes of the coarse grid coincide with the nodes of the fine grid, the displacement of the coarse grid node is assigned to the corresponding fine grid node; the midpoint on the edge of the coarse grid is the fine grid node, and the average of the sum of the displacements of the two nodes on the edge of the coarse grid is taken as the displacement value; the center point of the coarse grid is the fine grid node, and one fourth of the sum of the displacements of the four nodes of the coarse grid is taken as the displacement value, so as to obtain the displacement of the nine fine grid nodes contained in the coarse grid.

[0087] Step ten: using the node displacement q e and the stress function matrix S, the stress of the fine grid is calculated.

[0088] sigma = Sq e (8)

[0089] Step eleven: when the stress of the fine grid element exceeds the ultimate stress of the composite component, the fine grid element is damaged, and the fine grid region is regarded as a crack after the fine grid element is damaged, the corresponding fine grid material component is added to the air layer of the image, and the conservation of mass of the calculation model is maintained.

[0090] Step twelve: repeat steps seven to eleven until a connected crack is observed in the composite image, then stop the calculation process, and obtain the cracking of the composite material, as shown in Figure 2 .

[0091] The preferred embodiments of the application are described in detail above. It should be understood that those skilled in the art can make many modifications and changes without creative labor according to the concept of the present application. Therefore, any technical solution obtained by logical analysis, reasoning or limited experiment on the basis of the prior art according to the concept of the present application shall be within the protection scope determined by the claims.

Claims

1. A method for calculating the crack path of composite materials based on matrix-free pixel finite element method, characterized in that, Includes the following steps: Step 1: Acquire images of the composite material and perform preprocessing; Step 2: Establish a matrix-free pixel finite element fine mesh model; Step 3: Enlarge the size of the matrix-free pixel finite element fine mesh by one time to obtain the matrix-free pixel finite element coarse mesh model, and embed the boundary of the composite material into the unit to form an embedded boundary unit. Step 4: Based on the node coordinates of the matrix-free pixel finite element fine mesh, calculate the geometric matrix, Jacobian matrix, stiffness matrix of each material property fine mesh element, and stress function matrix of each material property fine mesh element. Step 5: Based on the dependency relationship between the coarse mesh element number and the fine mesh element number, calculate the number of the fine mesh element contained in the coarse mesh element, identify the material properties of the fine mesh element based on the number of the fine mesh element, calculate the stiffness matrix of the coarse mesh element based on the stiffness matrix obtained from the material properties of the fine mesh element, and use the averaging method to calculate the stiffness matrix of the coarse mesh element. Step 6: Using the element stiffness matrix of the coarse mesh, assemble the preprocessing matrix of the preprocessing conjugate gradient method using the diagonal matrix of its stiffness matrix. 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 using the stress function matrix, displacement load rate, number of cycles, and time step. Step 8: Calculate the displacement of the coarse mesh nodes using the preprocessed conjugate gradient method; Step 9: Calculate the displacement of the fine mesh nodes using interpolation based on the displacement of the coarse mesh nodes; Step 10: Calculate the stress in the fine mesh using the nodal displacement and stress function matrix of the fine mesh; Step 11: When the stress of the fine mesh element exceeds the ultimate stress of the composite material component, the fine mesh element fails. After the element fails, the element region is regarded as a crack. Corresponding material elements are added to the air layer of the image to maintain the material conservation of the calculation model. Repeat steps seven through eleven until a connected crack is observed in the composite material image, then stop the calculation process to obtain the crack path of the composite material.

2. The method for calculating the crack path of composite materials based on matrix-free pixel finite element method as described in claim 1, characterized in that, Step 1: Acquire composite material images and perform preprocessing. Specifically, this includes acquiring composite material images, segmenting the acquired composite material images, obtaining the distribution of each component of the composite material, and assigning a specific grayscale value to each component.

3. The method for calculating the crack path of composite materials based on matrix-free pixel finite element method as described in claim 2, characterized in that, An air layer is added to the boundary of the composite image, and a grayscale value of 175 is assigned.

4. The method for calculating the crack path of composite materials based on matrix-free pixel finite element method as described in claim 1, characterized in that, Step 2: Establish a matrix-free pixel finite element fine mesh model, which includes the number of pixels in the x-direction, the number of pixels in the y-direction, pixel grayscale values, material type, material properties, number of fine mesh nodes, number of fine mesh elements, degree of freedom, and element type.

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

6. The method for calculating the crack path of composite materials based on matrix-free pixel finite element method as described in claim 1, characterized in that, In step four, the Jacobian matrix J of the matrix-free pixel finite element fine mesh is calculated using Equation 1 based on the unit node coordinates of the matrix-free pixel finite element fine mesh. Where x and y are the coordinates in the physical coordinate system; ξ and η are the coordinates in the projected coordinate system.

7. The method for calculating the crack path of composite materials based on matrix-free pixel finite element method as described in claim 6, characterized in that, In step four, based on the element node coordinates of the matrix-free pixel finite element fine mesh, the geometric matrix B of the element in the matrix-free pixel finite element fine mesh is calculated using Equation 2. Where ξ and η are the coordinates of the projected coordinate system; J -1 N is the inverse of the Jacobian matrix; * (ξ,η) is the shape function matrix.

8. The method for calculating the crack path of composite materials based on matrix-free pixel finite element method as described in claim 7, characterized in that, In step four, based on the element node coordinates of the matrix-free pixel finite element fine mesh, the stiffness matrix K of the fine mesh element for each material property is determined. e Where t is the thickness of the plane problem, t = 1.

9. The method for calculating the crack path of composite materials based on matrix-free pixel finite element method as described in claim 8, characterized in that, In step four, based on the element node coordinates of the matrix-free pixel finite element fine mesh, the stress function matrix S of the fine mesh element for each material property is calculated using equation (4). S = CB (4) Where C is the elasticity coefficient matrix.

10. The method for calculating the crack path of composite materials based on matrix-free pixel finite element method as described in claim 8, characterized in that, In step five, based on the dependency relationship between coarse and fine mesh element numbers, the numbers of the fine mesh elements contained in the coarse mesh element are calculated. The material properties of the fine mesh elements are identified based on their numbers. The stiffness matrix of the coarse mesh element is calculated using the averaging method, specifically including the stiffness matrix K calculated based on the material properties of the fine mesh element. e The stiffness matrix of the coarse mesh element is calculated using equation (5). In the formula: J is the stiffness matrix of the fine-mesh element; i The constant Jacobian matrix is ​​calculated using the following formula: In the formula: h1 and h2 are the pixel dimensions in the x and y directions, respectively.

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