A Parallel Computation Method for Time-Domain Spectral Element Simulation of Ultrasonic Guided Wave Propagation Characteristics of CFRP Structures

Abstract

This invention discloses a parallel computation method for time-domain spectral element simulation of ultrasonic guided wave propagation in CFRP structures, relating to the field of nondestructive testing technology. The purpose of this invention is to address the low computational efficiency of current methods for ultrasonic guided wave propagation in large-area composite plate structures. It employs multi-node spectral elements to mesh the large-area composite plate, designs a mesh numbering algorithm to divide the mesh into sub-blocks (domain decomposition), creates subsets of disjoint nodes, and combines the high-order time-domain spectral element method with a Computational Unified Device Architecture (CUDA) to achieve parallel computation. A stepwise time integration algorithm based on a central difference scheme is used to implement unit-level parallel computation to solve the wave equation on a GPU. This method combines the high-order time-domain spectral element method (SEM) with CUDA, improving computational efficiency and accelerating the computation of ultrasonic guided wave propagation in large-area composite plate structures. It has broad application prospects in health monitoring methods for composite plate structures.

Description

TECHNICAL FIELD

[0001] The application relates to a CFRP structure ultrasonic guided wave propagation time domain spectral element simulation parallel computing method and belongs to the technical field of nondestructive testing. BACKGROUND

[0002] Ultrasonic guided wave detection is largely dependent on the propagation characteristics of waves in structures. Due to the complexity of the detection target, the diversity of the waves, and the complex interaction between different wave modes, the propagation problem of ultrasonic guided waves becomes particularly complex. In practical applications, factors such as the propagation path, frequency, wave speed, and attenuation of the waves are influenced by the structural material, geometry, boundary conditions, and other factors, making it difficult to accurately predict and analyze the behavior of the waves propagating in the structure. In particular, in complex structures such as composite materials, the propagation of the waves is influenced by factors such as anisotropy, interface effects, and damage, further increasing the difficulty of analyzing the propagation characteristics. Therefore, studying the propagation characteristics of ultrasonic guided waves, especially in complex structures, is crucial for improving the accuracy and reliability of ultrasonic guided wave detection.

[0003] Currently, numerical simulation is an effective means to study the propagation behavior of ultrasonic guided waves in structures. Traditionally, the classical finite element method (FEM) is used for modeling wave propagation problems, but since the finite element method uses linear elements, if high accuracy is required, dense meshes must be divided, resulting in long computation time, especially when dealing with large structures. In contrast, using high-order time domain spectral element method (SEM) combined with compute unified device architecture (CUDA) to realize parallelization has significant advantages. Spectral elements represented by high-order shape functions can achieve faster convergence speed under relatively coarse meshes compared to low-order finite elements. Parallel computing can significantly improve computational efficiency by using multiple processors to run simultaneously. GPUs have a rich structure of processing units and a wide range of memory hierarchies, providing powerful computing capabilities, especially for large-scale computing tasks. Therefore, GPU-based spectral element parallel computing not only accelerates the simulation process of ultrasonic guided wave propagation, but also effectively improves the simulation accuracy, which is of great significance for modeling ultrasonic guided wave propagation in large-area composite materials and can promote the research and application development in related fields. SUMMARY

[0004] The application proposes a CFRP structure ultrasonic guided wave propagation time domain spectral element simulation parallel computing method, aiming to realize parallel computing of ultrasonic guided wave propagation in CFRP laminate structures on GPUs.

[0005] In order to achieve the above technical purpose, the technical scheme of the application is as follows:

[0006] A CFRP structure ultrasonic guided wave propagation time domain spectral element simulation parallel computing method, characterized in that the method comprises:

[0007] Step 1: Use ABAQUS software to divide the composite material plate into a 4-node quadrilateral element mesh, and calculate the coordinates of the one-dimensional n-node Gauss-Lobatto-Legendre integration points. In an nth-order spectral element, the node coordinates are in the range [-1, 1]. The normalized coordinates of the nodes can be expressed as:

[0008] ξ1=-1,ξ2=r1,ξ3=r2,ξ n-1 =r n-2 ,ξ n =1

[0009] Where r1, r2, ..., r n-2 L represents the (n-2)th order Lobatto polynomial. n-2 The root of (ξ), the node ξ of an nth-order spectral unit. i The coordinates are obtained by the following formula:

[0010] (1-ξ 2 )L n-2 (ξ)=0

[0011] Where L n-2 (ξ) is a Lobatto polynomial:

[0012]

[0013] Where P n (ξ) is the set of first derivatives of (n+1)-order Legendre polynomials:

[0014]

[0015] The 4-node quadrilateral element is converted to an n-node quadrilateral element using the Gauss-Lobatto-Legendre integration point coordinates. 2 For nodal spectral elements, the coordinates of newly added nodes are obtained using the following formula:

[0016]

[0017] Where x is the global coordinate of the midpoint of the element edge, l is the element edge length, and ξ... i Let x be the coordinates of the integration point, and l be the intersection of the lines connecting the nodes on the edge of the relative element. In this case, x is the coordinate of the midpoint of the line connecting the two relative nodes, and l is the distance between the two relative nodes.

[0018] Step 2: Decompose the common nodes of the cells using a grid numbering algorithm, and assign local numbers to the separated cells. Considering four spectral cells with common nodes, when each spectral cell has 36 nodes, the correspondence between global node numbers and local node numbers is as follows:

[0019]

[0020] Where W is the global node number when spectral units are connected, and L is the local node number after domain decomposition. The superscript is the unit number, indicating which unit the node belongs to. A mapping is created connecting the global index W and the local index L. For example, local numbers 36, 67, 78, and 109 are paired with global number 61.

[0021]

[0022] The index pairs of global and local node numbers are used for assembling global force and displacement vectors:

[0023]

[0024] in and F represents the element nodal force matrix and element nodal displacement matrix obtained through parallel computation. W and U W These are the assembled global force matrix and global displacement matrix;

[0025] Step 3: Define the parameters required for the solution, including material properties, model geometry, excitation signal s(t), and time step Δt:

[0026]

[0027] Among them, f c The center frequency of the excitation signal is n, and the number of modulation cycles is n.

[0028] Step 4: Calculate the stiffness matrix components of the composite material based on the defined material parameters. The definitions of the delaminated and undelaminated regions are included in the stiffness matrix components. The stiffness matrix of the upper and lower layer elements in the delaminated region is calculated independently.

[0029]

[0030] Where n is the number of CFRP layers, ρ is the density, l is the total thickness of the shell, and a ij For tensile stiffness, b ij For bending-stretching coupling, d ij For bending stiffness, The stiffness tensor is calculated from the mechanical properties of a single layer of composite material and the rotation matrix of the material stacking sequence.

[0031] Step 5: Calculate the shape function S on the Gauss-Lobatto-Legendre nodes based on the coordinates of the Gauss-Lobatto-Legendre integration points. i The derivative of (x,y,z):

[0032]

[0033] in Let be the shape function of the i-th node of the element. Table (c) above represents the element shape functions. This represents element-wise multiplication, where ξ is the coordinate of the Gauss-Lobatto-Legendre integration point, and k(ξ) is the matrix k consisting of the coordinates of the integration points ξ. ξ (1:n,j)=ξ j-1 (j=1,...,n), where (1:n) represents all rows of the matrix, and n represents the number of nodes. The coefficients of the Vandermonde matrix The table above shows T as the transpose matrix, and the Jacobian matrix J is then calculated using shape functions. (b) :

[0034]

[0035] X (b) The coordinates of the cell nodes are shown below. The superscript (b) indicates that the matrix is ​​computed in parallel.

[0036] Step 6: Calculate strain for separate spectral units on the GPU using parallel methods. stress Internal force Where L is the index of the local node number, and the global force vector F is assembled by pairing the global node number created by the mesh numbering algorithm with the index of the local node number. W , where W is the index of the global node number, and then the displacement at time step t+Δt is calculated according to the stepwise time integration algorithm, where t represents the current time and Δt is the time step size;

[0037]

[0038] U t-Δt U t U t+Δt Let F represent the displacement at the previous time step, the displacement at the current time step, and the displacement to be solved at the next time step, respectively, where Δt is the time step and F is the displacement at the next time step. t Let M be the excitation at the current time t, M be the global mass matrix, and C be the global damping matrix;

[0039] Step 7: Visualize the full wavefield data by approximating the wavefield using nodal displacements and shape functions.

[0040] The advantages and beneficial effects of this invention are as follows:

[0041] This invention proposes a parallel computational method for time-domain spectral element simulation of ultrasonic guided wave propagation in CFRP structures. It employs multi-node spectral elements to mesh large-area composite plate panels, generating meshes typically with millions of degrees of freedom. Solving the dynamic propagation process of coupled guided waves requires significant computational resources. High-order shape functions, specifically Lagrange polynomials, are used for node interpolation, calculating the derivatives of local shape functions at Gauss-Lobatto-Legendre points. Sparse matrix storage is employed to improve computational efficiency. A mesh numbering algorithm is used to divide the mesh into sub-blocks (domain decomposition), and parallelization is achieved through a Computational Unified Device Architecture (CUDA). Element-level parallel computation is implemented on GPUs to solve the wave equations. Finally, a stepwise time integration algorithm is used to calculate the displacement at each time step. This parallel computational method for ultrasonic guided wave propagation modeling based on the time-domain spectral element method significantly improves computational efficiency, accelerating the simulation of ultrasonic guided wave propagation in large-area composite plate structures. It has broad engineering application value in composite material health monitoring and non-destructive testing. Attached Figure Description

[0042] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on the structures shown in these drawings without creative effort.

[0043] Figure 1 This is a flowchart of the method of the present invention;

[0044] Figure 2 This is a schematic diagram of the model mesh;

[0045] Figure 3 This is a diagram illustrating global and local node numbering.

[0046] Figure 4 This is a schematic diagram of wave propagation;

[0047] Figure 5 This is a schematic diagram of the defect signal received by the sensor;

[0048] Figure 6a This is a diagram illustrating CPU computation time.

[0049] Figure 6b This is a diagram illustrating GPU parallel computing time.

[0050] Figure 7 This is a schematic diagram of parallel computing acceleration; Detailed Implementation

[0051] To make the objectives, technical solutions, and advantages of the embodiments of the present invention clearer, the technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are some embodiments of the present invention, but not all embodiments. All other embodiments obtained by those skilled in the art based on the embodiments of the present invention without creative effort are within the scope of protection of the present invention.

[0052] ABAQUS is a powerful finite element method (FEM) software for engineering simulation. Its mesh generation function is a crucial part of FEM, decomposing large structures into smaller, simpler elements. By setting local mesh density, defining characteristic lines, and refining local regions, sufficient mesh density is ensured in critical areas to improve solution accuracy and efficiency. Gauss-Lobatto-Legendre is a numerical integration method used to calculate definite integrals over a given interval. Combining Gauss-Legendre integration points and endpoints (interval boundaries) provides higher accuracy during computation. A GPU (Graphics Processing Unit) is a processor specifically designed for processing graphics and images, widely used in parallel computing and other fields. Compared to CPUs, GPUs' parallel processing capabilities make them more efficient at handling large amounts of data, capable of processing thousands of threads in parallel. The Jacobian matrix is ​​primarily used to describe and process transformations from a global coordinate system to a local coordinate system; it represents the mapping relationship between coordinate systems. The Vandermonde matrix is ​​used for polynomial interpolation, node distribution, and the construction of element shape functions.

[0053] Example: such as Figure 1 As shown, this invention provides a parallel computation method for time-domain spectral element simulation of ultrasonic guided wave propagation in a CFRP structure:

[0054] 1) Step one of this method requires calculating the coordinates of the one-dimensional 6-node Gauss-Lobatto-Legendre integration points. In an nth-order spectral unit, the node coordinates are in the range [-1, 1]. The normalized coordinates of the nodes can be expressed as:

[0055] ξ1=-1,ξ2=r1,ξ3=r2,ξ n-1 =r n-2 ,ξ n =1

[0056] Where r1, r2, ..., r n-2 L represents the (n-2)th order Lobatto polynomial. n-2 The root of (ξ), the node ξ of an nth-order spectral unit. i The coordinates are obtained by the following formula:

[0057] (1-ξ 2 )L n-2 (ξ)=0

[0058] Where L n-2 (ξ) is a Lobatto polynomial:

[0059]

[0060] Where P n (ξ) is the set of first derivatives of (n+1)-order Legendre polynomials:

[0061]

[0062] The 4-node quadrilateral element is converted to an n-node quadrilateral element using the Gauss-Lobatto-Legendre integration point coordinates. 2 For nodal spectral elements, the coordinates of newly added nodes are obtained using the following formula:

[0063] X i =x + (l / 2)*ξ i

[0064] Where x is the global coordinate of the midpoint of the element edge, l is the element edge length, and ξ... i The coordinates of the integration point are given. The middle node of the spectral element is the intersection of the lines connecting the nodes on the edge of the relative element. In this case, x is the coordinate of the midpoint of the line connecting the two relative nodes, and l is the distance between the two relative nodes.

[0065] 2) In step two of this method, if Figure 2 As shown, the correspondence between global node numbers and local node numbers when considering four 36-node spectral elements is as follows:

[0066]

[0067]

[0068] Where W is the global node number when spectral units are connected, and L is the local node number after domain decomposition. The superscript is the unit number, indicating which unit the node belongs to, creating a mapping between the global index W and the local index L:

[0069]

[0070] The algorithm sets 12 sets of index pairs, so it can handle meshes with up to 12 cells connected to the same node. The sets of local-global index pairs have the same length, so the definition between all computational tasks ensures perfect load balancing.

[0071] 3) The material used in this example is 16 layers [45\0\-45\90\-45\0\45\90]. s A laminated symmetrical carbon fiber reinforced polymer (CFRP) composite board with a single layer thickness of 0.125 mm, a total thickness of 2 mm, and geometric dimensions of 500 mm × 500 mm × 2 mm.

[0072] 3) Calculate the overall stiffness matrix components and mass matrix components using the defined material parameters, dividing all nodes of the CFRP plate into non-delamination and delamination zones. In the non-delamination zone, calculate the stiffness components of each layer of material by cyclically indexing the node numbers; in the delamination zone, split the nodes into upper and lower layer numbers, and calculate the stiffness components of the upper and lower layers independently. The piezoelectric sensor is also considered a composite layer, which helps with the overall inertia of the component. The piezoelectric sensor is placed at the center of the plate. A delamination defect with a diameter of 5mm is placed 12mm from the center of the plate between the 4th and 5th layers of the composite material plate. The formulas for calculating the mass matrix components and stiffness matrix components are:

[0073]

[0074] Where n is the number of CFRP layers, ρ is the density, l is the total thickness of the shell, and a ij For tensile stiffness, b ij For bending-stretching coupling, d ij For bending stiffness, The stiffness tensor is calculated from the mechanical properties of a single layer of composite material and the rotation matrix of the material stacking sequence.

[0075] 4) After completing the material setup, perform preliminary calculations. First, calculate the shape function S on the Gauss-Lobatto-Legendre nodes based on the coordinates of the Gauss-Lobatto-Legendre integration points. i The derivative of (x,y,z):

[0076]

[0077] in Let be the shape function of the i-th node of the element. Table (c) above represents the element shape functions. This represents element-wise multiplication, where ξ is the coordinate of the Gauss-Lobatto-Legendre integration point, and k(ξ) is the matrix k consisting of the coordinates of the integration points ξ. ξ (:,j)=ξ j-1 (j=1,...,n): represents all rows, and n represents the number of nodes. The coefficients of the Vandermonde matrix The table above shows T as the transpose matrix, and the Jacobian matrix J is then calculated using shape functions. (b) :

[0078]

[0079] X (b) The coordinates of the unit node are shown in the superscript (b), indicating that the matrix is ​​computed in parallel.

[0080] 5) Calculate the Jacobian matrix J based on the derivatives of the shape functions. (b) ;

[0081]

[0082] Where X (b) Y (b) and Z (b) These are the global coordinates of the cell nodes. The superscript (b) indicates that the matrix is ​​computed in parallel. and The derivative matrix of the global shape functions is derived from the derivative matrix of the unit shape functions. and The copied matrix, in block diagonal form, is very sparse; in spectral units with interpolation orders a = b = 5 and c = 2, the number of non-zero elements is less than 5%.

[0083]

[0084] 6) Stepwise time integration algorithm based on the central difference scheme:

[0085]

[0086] Where M is the global mass matrix, K is the global stiffness matrix, C is the global damping matrix, U is the vector of global degrees of freedom, and F... t The vector of transient excitation, and The expression is:

[0087]

[0088] Will and Substituting the expression for the stepwise time integration algorithm, we get:

[0089]

[0090] Where F W =KU,F t It is a transient excitation;

[0091] 7) Parallel methods are used to assemble internal force vectors for separated spectral elements. The key element of the parallel method is sparse matrix-vector multiplication. Instead of assembling the global stiffness matrix, it performs stiffness component and nodal displacement calculations. The product of these components is used to convert the resulting data into a GPU array for accelerated computation using GPU. The first step is to calculate the strain components at time t. stress Internal force

[0092]

[0093] in The second step involves element-wise multiplication within the matrix, followed by calculation of nodal stresses.

[0094]

[0095] The next step is to calculate the effective global forces along the x, y, and z directions respectively:

[0096]

[0097] Assemble the global force vector F based on the index pairs of the grid numbering algorithm. W :

[0098]

[0099] The displacement at time step t+Δt is calculated using the stepwise time integration algorithm expression:

[0100]

[0101] U t-Δt U t U t+Δt Let F represent the displacement at the previous time step, the displacement at the current time step, and the displacement to be solved at the next time step, respectively, where Δt is the time step and F is the displacement at the next time step. t The excitation is at the current time t, M is the global mass matrix, C is the global damping matrix, and after completion, the displacement data of all nodes are saved at each time step.

[0102] 8) The signal received by the piezoelectric sensor in the model is realized by the following formula:

[0103]

[0104] In the table above, Pzt represents the piezoelectric sensor node, and h is the CFRP plate thickness. pzt Where is the thickness of the piezoelectric sensor, e is the piezoelectric coefficient, and q is the piezoelectric constant;

[0105] 9) After obtaining all nodal displacement data, a uniform grid interpolation technique is used to visualize the full wavefield data. First, a uniform data point (xp, yp) is created. Then, the spectral cell containing the data point is identified. The spectral node (x0, y0) within that cell that is closest to the data point (xp, yp) is used as the initial point. Parallel computation is then performed on all data points within the regular grid.

[0106]

[0107] In the first iteration, k = 0;

[0108] 10) Next, calculate the shape function and Jacobian matrix at the point (ξ1,η1), and use the following formula to calculate the global coordinates (x1,y1) of the interpolation point corresponding to the local coordinates (ξ1,η1):

[0109]

[0110] 11) After 3 iterations, the new interpolated local coordinates (ξ) are obtained. p ,η p ), in the interpolated local coordinates (ξ) p ,η p The following calculations are performed on the shape functions of the spectral unit:

[0111] S (c) =(Q) T *k

[0112] Where k(i,j)=ξ i-1 *η j-1 (i = 1, ..., 6, j = 1, ..., 6), where Q is the coefficient of the Vandermonde matrix;

[0113] 12) Extract the displacement data of all nodes of the spectral element containing the data point (xp, yp);

[0114] 13) Approximating the wave field using nodal displacements and shape functions:

[0115]

[0116] The purpose of this invention is to address the low computational efficiency of ultrasonic guided wave propagation calculations for large-area composite plate structures. The aim is to achieve parallel computation of ultrasonic guided wave propagation in large-area composite plate structures, thereby improving computational efficiency. The method described in this invention combines the high-order time-domain spectral element method with the Computational Unified Device Architecture (CUDA) to achieve parallelization. It uses multi-node spectral elements to model large-area composite plates, decomposes the mesh domain into local elements using mesh numbering technology, and then performs parallel computation on a GPU through a stepwise time integration algorithm, thus accelerating the computation of ultrasonic guided wave propagation in composite plates.

[0117] The above description is a further detailed explanation of the present invention in conjunction with specific preferred embodiments. It should not be considered that the specific implementation of the present invention is limited to these descriptions. For those skilled in the art, several simple deductions or substitutions can be made without departing from the concept of the present invention, and all such deductions or substitutions should be considered to fall within the protection scope of the present invention.

Claims

1. A method for parallel computation of time-domain spectral element mode analog of ultrasonic guided wave propagation in CFRP structures, characterized in that, The method comprises the following steps: Step one: using ABAQUS software to divide the composite material plate into 4-node quadrilateral element grid, calculate one-dimensional n-node Gauss-Lobatto-Legendre integral point coordinates, and convert 4-node quadrilateral element into n 2 node spectral element through Gauss-Lobatto-Legendre integral point coordinates ; Step two: using grid numbering algorithm to decompose the unit common node, local numbering for the separated unit, creating the corresponding relationship between global node number W and local node number L, used for the index of force and displacement vector; Step three: define the parameters needed to solve, including material properties, geometry of the model, and excitation signal S(t) and time step ; Step four: calculating the composite material mass matrix component and stiffness matrix component according to the defined material parameters, wherein the layered area and the unlayered area are defined in the stiffness matrix component, and the upper and lower layer units of the layered area are independently calculated for the stiffness matrix component; Step five: Calculate the three-dimensional shape function on Gauss-Lobatto-Legendre nodes according to the Gauss-Lobatto-Legendre integral point coordinates and the integral weight W (b) , and calculate the Jacobian matrix J according to the three-dimensional shape function (b) , where i represents the node number, the superscript (b) indicates that the matrix adopts parallel computing, the three-dimensional shape function is obtained by the tensor product of the one-dimensional shape functions S(x), S(y), S(z) on the x, y, z coordinates; Step six: Compute strain on GPU for isolated spectral element using parallel approach , stress , i,j = 1,2,3, internal force where L is the index of local node numbering, the global node numbering is assembled from the index of local node numbering and global node numbering according to the mesh numbering algorithm where W is the index of global node numbering, the displacement at time step is calculated according to the step-by-step time integration algorithm, where t represents the current time, is the time step; Step seven: visualizing the full wave field data, and using the node displacement and shape function to approximate the wave field.

2. The method according to claim 1, wherein, In step one, the one-dimensional n-node Gauss-Lobatto-Legendre integral point coordinates are calculated, and the node coordinates are in the range of [-1, 1] in an n-order spectral element, so the normalized coordinates of the nodes can be expressed as: where denotes the roots of the (n-2)th Lobatto polynomial of a spectral element of order n The coordinates of the nodes of a spectral element of order n are given by wherein is the Lobatto polynomial: wherein a set of first derivatives of Legendre polynomials of order (n + 1): The 4-node quadrilateral element is converted into n 2 node spectral element by Gauss-Lobatto-Legendre integration point coordinates. The new node coordinates are obtained by the following formula: where x is the global coordinate of the midpoint of the edge, is the length of the edge, is the coordinate of the integration point, the midpoint of the spectral element is the intersection of the lines connecting the nodes on the relative edge, and x is the coordinate of the midpoint of the line connecting the two nodes, is the distance between the two nodes.

3. The parallel computing method for time-domain spectral element mode simulation of ultrasonic guided wave propagation characteristics of CFRP structures according to claim 1, characterized in that, In step two, considering four spectral elements with common nodes, if each spectral element has 36 nodes, the corresponding relationship between the global node number and the local node number is as follows: Wherein W is the global node number when the spectral elements are connected, L is the local node number after the domain decomposition, the superscript is the element number, indicating which element the node belongs to, and the mapping connecting the global index W and the local index L is created, and the index pair is used for assembling the global force or displacement vector after parallel computing: wherein and are the element node force matrix and element node displacement matrix obtained by parallel computation, and are the assembled global force matrix and global displacement matrix.

4. The parallel computing method for the time-domain spectral element mode simulation of ultrasonic guided wave propagation in CFRP structures according to claim 1, characterized in that, In step four, the layered area and the unlayered area are defined in the stiffness matrix component, and the upper and lower layer units of the layered area are independently calculated for the stiffness matrix: where n is the number of CFRP plies, is the density, is the total thickness of the shell, is the tensile stiffness, is the bending-tensile coupling, is the bending stiffness, is the stiffness tensor, calculated from the mechanical properties of the composite material single ply and the rotation matrix of the material stacking sequence.

5. The parallel computing method for time-domain spectral element mode simulation of ultrasonic guided wave propagation characteristics of CFRP structures according to claim 1, characterized in that, In step five, the derivatives of the shape functions on the nodes are computed: where is the shape function of the i-th node of the element, the superscript (c) denotes the element shape function, denotes the element level multiplication, is the Gauss-Lobatto-Legendre integration point coordinate, is the integration point coordinate is the matrix composed of where (1:n) denotes all rows of the matrix and n denotes the number of nodes, is the coefficient of the Vandermonde matrix the superscript T denotes the transposed matrix, and the Jacobian matrix J is calculated from the shape functions (b) : X (b) Global coordinates of the unit node, superscript (b) indicates that the matrix adopts parallel computing mode.

6. The parallel computing method for the time-domain spectral element mode simulation of ultrasonic guided wave propagation in CFRP structures according to claim 1, characterized in that, In step six, the displacement at time step is calculated after assembly of the global force vector is completed: wherein , , respectively represent the displacement at the previous time, the displacement at the current time and the displacement at the next time to be solved, is the time step, is the excitation at the current time t, M is the global mass matrix, and C is the global damping matrix.

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