Numerical calculation method for simulating two-dimensional space crack propagation and related device

Abstract

The invention discloses a numerical calculation method for simulating two-dimensional space crack propagation and a related device, and relates to the technical field of computational mechanics, and the method comprises the following steps: obtaining an overall crack solving region and a crack tip region of a cantilever beam; establishing a transition area based on the crack overall area and the crack tip area; constructing a transition region basis function based on the transition region; based on the transition area primary function, the overall area primary function and the crack tip area primary function, solving the transition area, the overall area and the crack tip area to obtain a two-dimensional crack propagation result; the overall area primary function is a finite element primary function obtained after linear quadrilateral or triangular unit discretization; the primary function of the crack tip area is a B spline primary function; the two-dimensional crack propagation result comprises an overall displacement solution, a crack tip area displacement solution and a transition area displacement solution. According to the method, the numerical value of two-dimensional crack propagation can be accurately and stably simulated.

Description

Technical Field

[0001] The present application relates to the technical field of computational mechanics, and particularly to a numerical calculation method and related device for simulating crack propagation in a two-dimensional space. Background Art

[0002] Numerical simulation of crack problems has always been a research hotspot in the field of computational mechanics. The simulation of crack propagation has high requirements for numerical simulation accuracy and stability. The stress field distribution at the crack tip is complex, and the calculation of the crack propagation process generally requires mesh redivision, making it difficult to simultaneously meet the high accuracy and high stability of the simulation process. Therefore, the simulation effect of most numerical methods for crack propagation is not ideal. Summary of the Invention

[0003] The purpose of the present application is to provide a numerical calculation method and related device for simulating crack propagation in a two-dimensional space, which can accurately and stably simulate the numerical value of two-dimensional crack propagation.

[0004] To achieve the above purpose, the present application provides the following solutions.

[0005] In the first aspect, the present application provides a numerical calculation method for simulating crack propagation in a two-dimensional space, and the numerical calculation method for simulating crack propagation in a two-dimensional space includes the following steps.

[0006] Obtain the overall solution region and the crack tip region of the cantilever beam with a crack.

[0007] Based on the overall crack region and the crack tip region, establish a transition region.

[0008] Construct a basis function for the transition region based on the transition region.

[0009] Solve the transition region, the overall region, and the crack tip region based on the basis function of the transition region, the basis function of the overall region, and the basis function of the crack tip region to obtain the two-dimensional crack propagation result; the basis function of the overall region is a finite element basis function obtained by discretization using linear quadrilateral or triangular elements; the basis function of the crack tip region is a B-spline basis function; the two-dimensional crack propagation result includes: the overall displacement solution, the displacement solution of the crack tip region, and the displacement solution of the transition region.

[0010] In the second aspect, the present application provides a computer device, including: a memory, a processor, and a computer program stored on the memory and executable on the processor, and the processor executes the computer program to implement the above-mentioned numerical calculation method for simulating crack propagation in a two-dimensional space.

[0011] In a third aspect, the present application provides a computer-readable storage medium having a computer program stored thereon, which, when executed by a processor, implements the above-mentioned numerical calculation method for simulating crack extension in two-dimensional space.

[0012] According to the specific embodiments provided in this application, this application discloses the following technical effects.

[0013] The present application provides a numerical calculation method and related device for simulating crack propagation in two-dimensional space. First, the overall solution area and crack tip area of ​​the cantilever beam are obtained, which can accurately define the spatial range that needs to be numerically calculated. The overall solution area covers most of the cantilever beam except the crack tip, while the crack tip area focuses on the parts where stress and strain change most drastically. This division enables the subsequent numerical calculation to adopt different calculation strategies for different areas in a targeted manner, avoiding the use of a unified and overly complex calculation model in the entire cantilever beam range, thereby improving the calculation efficiency. Secondly, based on the overall crack area and the crack tip area, a transition area is established, which can effectively alleviate the numerical solution difference between the overall area and the crack tip area. By establishing a transition area, the overall area, the crack tip area and the subsequent constructed transition area basis functions are organically linked. This makes the crack solution model of the entire cantilever beam more complete in mathematical and physical senses, and the various areas are coordinated with each other, which can more realistically reflect the expansion of the crack in the cantilever beam, rather than a simple patchwork after the independent solution of each area. Then, based on the transition region, the transition region basis function is constructed to ensure that the basis function transition between different regions is continuous and smooth, avoiding numerical oscillation or error accumulation caused by sudden changes in the basis function, thereby making the entire solution process more stable and the results obtained more reliable. Finally, based on the transition region basis function, the overall region basis function and the crack tip region basis function, the transition region, the overall region and the crack tip region are solved to obtain two-dimensional crack propagation results. These detailed solution results provide comprehensive and accurate data support for subsequent structural safety assessments, fatigue life predictions and other analyses, helping engineers to better evaluate the performance and remaining service life of cantilever beams. BRIEF DESCRIPTION OF THE DRAWINGS

[0014] In order to more clearly illustrate the embodiments of the present application or the technical solutions in the prior art, the drawings required for use in the embodiments will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present application. For ordinary technicians in this field, other drawings can be obtained based on these drawings without paying creative work.

[0015] Figure 1 This is a diagram of the application environment of a numerical calculation method for simulating crack propagation in two-dimensional space in one embodiment of the present application.

[0016] Figure 2 Schematic flow chart of a numerical calculation method for simulating crack propagation in a two-dimensional space provided by an embodiment of the present application.

[0017] Figure 3 Schematic diagram of a circular region at the crack tip provided by an embodiment of the present application.

[0018] Figure 4 Schematic diagram of the transition region between the whole and the circular region provided by an embodiment of the present application.

[0019] Figure 5 Schematic diagram of a calculation model of a cracked cantilever beam provided by an embodiment of the present application.

[0020] Figure 6a Schematic diagram of a mesh model 1 of the whole and the crack tip region of a cracked cantilever beam provided by an embodiment of the present application.

[0021] Figure 6b Schematic diagram of a mesh model 2 of the whole and the crack tip region of a cracked cantilever beam provided by an embodiment of the present application.

[0022] Figure 7 Schematic diagram of a crack propagation path provided by an embodiment of the present application.

[0023] Figure 8 Schematic diagram of the structure of a computer device provided by an embodiment of the present application. Detailed implementation manners

[0024] Next, the technical solutions in the embodiments of the present application will be clearly and completely described in conjunction with the accompanying drawings in the embodiments of the present application. Obviously, the described embodiments are only a part of the embodiments of the present application, rather than all the embodiments. Based on the embodiments in the present application, all other embodiments obtained by those of ordinary skill in the art without creative efforts shall fall within the protection scope of the present application.

[0025] To make the above objects, features, and advantages of the present application more obvious and understandable, the present application will be further described in detail below in conjunction with the drawings and specific implementation manners.

[0026] The numerical calculation method for simulating crack propagation in a two-dimensional space provided by the embodiments of the present application can be applied to, for example Figure 1In the application environment shown. Among them, the terminal 102 communicates with the server 104 through the network. The data storage system can store the data that the server 104 needs to process. The data storage system can be set separately, integrated on the server 104, placed on the cloud or other servers. The terminal 102 can send the obtained overall solution area and crack tip area of the cantilever beam crack to the server 104. After receiving the overall solution area and crack tip area of the cantilever beam crack, for the overall solution area and crack tip area of the cantilever beam, the server 104 establishes a transition area based on the overall crack area and the crack tip area; constructs a transition area basis function based on the transition area; solves the transition area, overall area and crack tip area based on the transition area basis function, overall area basis function and crack tip area basis function to obtain a two-dimensional crack propagation result; the overall area basis function is a finite element basis function obtained by discretization using linear quadrilateral or triangular elements; the crack tip area basis function is a B-spline basis function; the two-dimensional crack propagation result includes: overall displacement solution, crack tip area displacement solution and transition area displacement solution. The server 104 can feedback the obtained two-dimensional crack propagation result to the terminal 102. In addition, in some embodiments, the numerical calculation method for simulating two-dimensional space crack propagation can also be implemented separately by the server 104 or the terminal 102. For example, the terminal 102 can directly perform numerical calculation of simulating two-dimensional space crack propagation for the overall solution area and crack tip area of the cantilever beam crack, or the server 104 can obtain the overall solution area and crack tip area of the cantilever beam crack from the data storage system and perform numerical calculation of simulating two-dimensional space crack propagation for the overall solution area and crack tip area of the cantilever beam crack.

[0027] Among them, the terminal 102 can be, but is not limited to, various desktop computers, laptop computers, smart phones and tablet computers. The server 104 can be implemented by an independent server or a server cluster composed of multiple servers, and can also be a cloud server.

[0028] In an exemplary embodiment, as Figure 2 shown, a numerical calculation method for simulating two-dimensional space crack propagation is provided. This method is executed by a computer device, and can be specifically executed separately by a computer device such as a terminal or a server, or jointly executed by a terminal and a server. In the embodiment of the present application, taking this method applied to Figure 1 the server 104 in as an example for illustration, it includes the following steps S1 to S4.

[0029] S1: Obtain the overall solution area and crack tip area of the cantilever beam crack.

[0030] S2: Establish a transition region based on the overall crack region and the crack tip region.

[0031] S3: Construct a basis function for the transition region based on the transition region.

[0032] S4: Solve for the transition region, the overall region, and the crack tip region based on the basis function of the transition region, the basis function of the overall region, and the basis function of the crack tip region to obtain the two-dimensional crack propagation result; the basis function of the overall region is the finite element basis function obtained by discretizing using linear quadrilateral or triangular elements; the basis function of the crack tip region is the B-spline basis function; the two-dimensional crack propagation result includes: the overall displacement solution, the displacement solution in the crack tip region, and the displacement solution in the transition region.

[0033] Implementing the above steps S1 to S4, first consider discretizing the overall solution region using a finite element mesh, and establishing a local coordinate system with the crack tip as the center in the crack tip region, and using parametric description for the region geometry and the crack within the region. Establish a transition region based on the circular region boundary and construct relevant basis functions. Use B-spline basis functions to approximate the corresponding circular curve and the transition region. In the transition region, introduce relevant weight functions to transform the finite element basis function and the circular region basis function in the transition region. The transformed relevant basis functions can at least reconstruct linear polynomials to ensure the numerical accuracy of the transition region and the stability of the overall solution. Compared with the existing methods, the present application can establish a local coordinate system at the crack tip to simulate the stress field in the crack tip region, accurately capture the singular characteristics of the crack solution, and has very good numerical solution stability; at the same time, a new coupling technology is used to seamlessly connect the basis function of the crack tip region and the overall finite element basis function, making the calculation during the crack propagation process maintain high efficiency and stability.

[0034] Discretize the overall finite element mesh.

[0035] If the overall solution region is discretized using linear quadrilateral or triangular elements, the overall displacement solution can be expressed by the following formula.

[0036] (1).

[0037] Where, is the overall displacement solution; is the finite element basis function of the overall region, is the number of finite element basis functions in the overall region; is the first variable to be solved for.

[0038] The expression of the overall displacement solution in the two-dimensional space can be expressed by the following formula.

[0039] (2).

[0040] (3).

[0041] Among them, is the overall displacement solution in the x - direction of the two - dimensional space; is the overall displacement solution in the y - direction of the two - dimensional space; is the finite - element basis function in the two - dimensional space; is the first variable to be solved in the x - direction of the two - dimensional space; is the first variable to be solved in the y - direction of the two - dimensional space.

[0042] B - spline approximation in the crack - tip region.

[0043] The displacement solution in the crack - tip region is approximated by the B - spline basis function.

[0044] (4).

[0045] Among them, is the displacement solution in the crack - tip region; is the B - spline basis function in the crack - tip region, is the number of B - spline basis functions in the crack - tip region; is the second variable to be solved; is the parameter - space variable.

[0046] The expression of the displacement solution in the crack - tip region in the two - dimensional parameter space can be represented by the following formula.

[0047] (5).

[0048] (6).

[0049] (7).

[0050] Among them, is the displacement solution in the x - direction of the crack - tip region in the two - dimensional parameter space; is the displacement solution in the y - direction of the crack - tip region in the two - dimensional parameter space; is the B - spline basis function of the crack - tip region in the two - dimensional parameter space; is the second variable to be solved in the x - direction of the two - dimensional parameter space; is the second variable to be solved in the y - direction of the two - dimensional parameter space; is the circumferential one - dimensional B - spline basis function; is the radial one - dimensional B - spline basis function; is the two - dimensional parameter - space coordinate.

[0051] Transition - region coupling technology.

[0052] Such as Figure 3 andFigure 4 As shown, the circular radius of the boundary of the circular region centered at the crack tip is , then a circle centered at the crack tip with a radius of and a circle with a radius of enclose a transition region . The outer and inner circular boundaries of the transition region are approximately represented by B-spline curves.

[0053] (8).

[0054] (9).

[0055] Among them, is the outer circular boundary curve of the transition region; is the inner circular boundary curve of the transition region; is the control point of the B-spline basis function corresponding to the outer circular boundary curve; is the control point of the B-spline basis function corresponding to the inner circular boundary curve; is the B-spline basis function; is the number of B-spline basis functions in the transition region.

[0056] In a two-dimensional space, the physical coordinates of a curve can be expressed by the following formula.

[0057] (10).

[0058] (11).

[0059] (12).

[0060] (13).

[0061] (14).

[0062] Among them, is the coordinate of the control point in the two-dimensional space; is the coordinate of the control point in the two-dimensional space; ([[]] , ) is the physical coordinate of the control point of the B-spline basis function corresponding to the outer circular boundary curve of the transition region; ([[]] , ) is the physical coordinate of the control point of the B-spline basis function corresponding to the inner circular boundary curve of the transition region.

[0063] Then the expression of the transition region in the two-dimensional parameter space is as follows.

[0064] (15).

[0065] (16).

[0066] (17).

[0067] Among them, ( , ) is the representation of the transition region in the two-dimensional parameter space.

[0068] The expression of the basis function of the transition region is as follows.

[0069] (18).

[0070] (19).

[0071] (20).

[0072] Among them, is the B-spline basis function of the transition region in the two-dimensional parameter space; is the finite element basis function of the transition region in the two-dimensional parameter space; is the B-spline basis function of the crack tip region in the two-dimensional parameter space; is the finite element basis function of the overall solution region in the two-dimensional parameter space; is the basic weight function; is the two-dimensional parameter space coordinate.

[0073] This application also provides an application scenario, which applies the above numerical calculation method for simulating crack propagation in a two-dimensional space. Specifically: The numerical calculation method for simulating crack propagation in a two-dimensional space provided in this embodiment can be applied in a crack propagation simulation scenario. The crack propagation simulation scenario includes: a data acquisition link, a transition region establishment link, a transition region basis function construction link, and a two-dimensional crack propagation link; First, obtain the overall solution region and the crack tip region of the crack of the cantilever beam; Secondly, based on the overall crack region and the crack tip region, establish a transition region; Then, based on the transition region, construct a transition region basis function; Finally, based on the transition region basis function, the overall region basis function, and the crack tip region basis function, solve the transition region, the overall region, and the crack tip region to obtain the two-dimensional crack propagation result; The overall region basis function is the finite element basis function obtained after discretization using linear quadrilateral or triangular elements; The crack tip region basis function is the B-spline basis function; The two-dimensional crack propagation result includes: the overall displacement solution, the crack tip region displacement solution, and the transition region displacement solution.

[0074] The following uses an actual example to verify the effectiveness of this application. AsFigure 5 As shown, there is a pre-existing crack in the two-dimensional cantilever beam. The left end of the beam is subjected to concentrated forces in opposite directions. The overall finite element analysis uses structural meshes. The crack tip region for local singularity and other geometric analyses is set as a circular region centered at the crack tip, as Figure 6a and Figure 6b shown. Two different-sized mesh models are used for the simulation analysis of crack propagation.

[0075] In Figure 6a , the finite element mesh , the radius of the circular region , and the B-spline nodes for analysis within the crack tip region are .

[0076] In Figure 6b , the finite element mesh , the radius of the circular region , and the B-spline nodes for analysis within the crack tip region are .

[0077] The calculation of the crack propagation direction is based on the maximum circumferential stress criterion. Two expansion step sizes, 1 and 0.5 respectively, are used, and the stress intensity factor is calculated using the interactive integral method. Figure 7 Showing the crack propagation path, it can be seen that with the same expansion step size, the crack propagation paths calculated based on the two mesh models are almost the same. The differences in the crack propagation paths under different expansion step sizes are obvious, and the simulated crack propagation of this application has good stability.

[0078] In an exemplary embodiment, a computer device is provided. The computer device can be a server or a terminal, and its internal structure diagram can be as Figure 8 shown. The computer device includes a processor, a memory, an input / output interface (Input / Output, abbreviated as I / O), and a communication interface. Among them, the processor, the memory, and the input / output interface are connected through a system bus, and the communication interface is connected to the system bus through the input / output interface. Among them, the processor of the computer device is used to provide computing and control capabilities. The memory of the computer device includes a non-volatile storage medium and an internal memory. The non-volatile storage medium stores an operating system, a computer program, and a database. The internal memory provides an environment for the operation of the operating system and the computer program in the non-volatile storage medium. The database of the computer device is used to store the overall solution region and the crack tip region of the cantilever beam. The input / output interface of the computer device is used to exchange information between the processor and external devices. The communication interface of the computer device is used to communicate with external terminals through a network connection. When the computer program is executed by the processor, it implements a numerical calculation method for simulating crack propagation in two-dimensional space.

[0079] Those skilled in the art can understand that Figure 8 the structure shown in Figure 8 is only a block diagram of some structures related to the solution of this application, and does not constitute a limitation on the computer device to which the solution of this application is applied. The specific computer device may include more or fewer components than those shown in the figure, or combine some components, or have a different component layout.

[0080] In an exemplary embodiment, a computer device is further provided, including a memory and a processor. A computer program is stored in the memory, and when the processor executes the computer program, the above-mentioned method embodiments are implemented.

[0081] In an exemplary embodiment, a computer-readable storage medium is provided, storing a computer program, and when the computer program is executed by a processor, the above-mentioned method embodiments are implemented.

[0082] It should be noted that the user information (including but not limited to user device information, user personal information, etc.) and data (including but not limited to data for analysis, stored data, displayed data, etc.) involved in this application are all information and data authorized by the user or fully authorized by all parties, and the collection, use, and processing of relevant data need to comply with relevant regulations.

[0083] Those of ordinary skill in the art can understand that all or part of the processes in the methods of the above embodiments can be completed by instructing relevant hardware through a computer program. The computer program can be stored in a non-volatile computer-readable storage medium. When the computer program is executed, it can include the processes of the embodiments of the above methods. Among them, any reference to a memory, database, or other medium used in the embodiments provided in the present application can include at least one of non-volatile and volatile memories. Non-volatile memory can include read-only memory (ROM), magnetic tape, floppy disk, flash memory, optical memory, high-density embedded non-volatile memory, resistive random access memory (ReRAM), magnetoresistive random access memory (MRAM), ferroelectric random access memory (FRAM), phase change memory (PCM), graphene memory, etc. Volatile memory can include random access memory (RAM) or external cache memory, etc. By way of illustration and not limitation, RAM can be in various forms, such as static random access memory (SRAM) or dynamic random access memory (DRAM), etc.

[0084] The databases involved in the embodiments provided in the present application can include at least one of relational databases and non-relational databases. Non-relational databases can include distributed databases based on blockchain, etc., without limitation. The processors involved in the embodiments provided in the present application can be general-purpose processors, central processors, graphics processors, digital signal processors, programmable logic devices, data processing logics based on quantum computing, etc., without limitation.

[0085] The technical features of the above embodiments can be combined arbitrarily. For the sake of brevity of description, not all possible combinations of the technical features in the above embodiments are described. However, as long as there is no contradiction in the combination of these technical features, it should be considered as the scope described in this specification.

[0086] Specific examples are used in this article to elaborate on the principles and implementation manners of the present application. The descriptions of the above embodiments are only used to help understand the methods and core ideas of the present application; at the same time, for those of ordinary skill in the art, according to the ideas of the present application, there will be changes in the specific implementation manners and application scopes. In summary, the content of this specification should not be construed as a limitation to the present application.

Claims

1. A numerical calculation method for simulating crack propagation in a two-dimensional space, characterized in that, The numerical calculation method for simulating two-dimensional crack propagation includes: Obtaining the overall solution region and crack tip region of the cantilever beam; Based on the overall crack region and the crack tip region, establishing a transition region; Constructing a basis function for the transition region based on the transition region; Solving the transition region, overall region, and crack tip region based on the basis function of the transition region, basis function of the overall region, and basis function of the crack tip region to obtain the two-dimensional crack propagation result; the basis function of the overall region is a finite element basis function obtained by discretizing using linear quadrilateral or triangular elements; the basis function of the crack tip region is a B-spline basis function; the two-dimensional crack propagation result includes: overall displacement solution, crack tip region displacement solution, and transition region displacement solution.

2. The numerical calculation method for simulating two-dimensional spatial crack propagation according to claim 1, characterized in that, The expression of the basis function of the transition region is: ； ； ； Among them, is the B-spline basis function of the transition region in the two-dimensional parameter space; is the finite element basis function of the transition region in the two-dimensional parameter space; is the B-spline basis function of the crack tip region in the two-dimensional parameter space; is the finite element basis function of the overall solution region in the two-dimensional parameter space; is the basic weight function; is the two-dimensional parameter space coordinate.

3. The numerical calculation method for simulating two-dimensional spatial crack propagation according to claim 1, characterized in that The expression of the overall displacement solution is: ； Among them, is the overall displacement solution; is the finite element basis function of the overall region, is the number of finite element basis functions in the overall region; is the first variable to be solved for.

4. The numerical calculation method for simulating two-dimensional spatial crack propagation according to claim 1, characterized in that, The expression of the overall displacement solution in two-dimensional space is: ； ； Among them, is the overall displacement solution in the x-direction of the two-dimensional space; is the overall displacement solution in the y-direction of the two-dimensional space; is the finite element basis function in the two-dimensional space; is the first variable to be solved in the x-direction of the two-dimensional space; is the first variable to be solved in the y-direction of the two-dimensional space.

5. The numerical calculation method for simulating two-dimensional spatial crack propagation according to claim 1, characterized in that The expression of the crack tip region displacement solution is: ； Among them, is the displacement solution in the crack tip region; is the B-spline basis function in the crack tip region, is the number of B-spline basis functions in the crack tip region; is the second variable to be solved.

6. The numerical calculation method for simulating two-dimensional spatial crack propagation according to claim 1, characterized in that, The expression of the crack tip region displacement solution in two-dimensional parameter space is: ； ； ； Among them, is the displacement solution in the crack tip region in the x - direction of the two - dimensional parameter space; is the displacement solution in the crack tip region in the y - direction of the two - dimensional parameter space; is the B - spline basis function in the crack tip region in the two - dimensional parameter space; is the second variable to be solved in the x - direction of the two - dimensional parameter space; is the second variable to be solved in the y - direction of the two - dimensional parameter space; is the circumferential one - dimensional B - spline basis function; is the radial one - dimensional B - spline basis function; is the coordinate in the two - dimensional parameter space.

7. The numerical calculation method for simulating two-dimensional spatial crack propagation according to claim 1, wherein, The expression of the boundary curve of the transition region is: ； ； ； ； ； ； ； Among them, is the circular boundary curve of the periphery of the transition region; is the circular boundary curve inside the transition region; is the control point of the B-spline basis function corresponding to the circular boundary curve of the periphery; is the control point of the B-spline basis function corresponding to the circular boundary curve inside; is the B-spline basis function; is the number of B-spline basis functions in the transition region; is the control point in the coordinates of the two-dimensional space; is the control point in the coordinates of the two-dimensional space; ( , ) are the physical coordinates of the control points of the B-spline basis function corresponding to the circular boundary curve of the periphery of the transition region; ( , ) are the physical coordinates of the control points of the B-spline basis function corresponding to the circular boundary curve inside the transition region.

8. The numerical calculation method for simulating two-dimensional spatial crack propagation according to claim 1, wherein, The expression of the transition region in two-dimensional parameter space is: ； ； ； Among them, ( , ) are the physical coordinates of the control points of the B-spline basis functions corresponding to the circular boundary curve of the periphery of the transition region; ( , ) are the physical coordinates of the control points of the B-spline basis functions corresponding to the circular boundary curve inside the transition region; is the two-dimensional parameter space coordinate; ( , ) is the representation of the transition region in the two-dimensional parameter space.

9. A computer device, comprising: A memory, a processor, and a computer program stored on the memory and executable on the processor, wherein the processor executes the computer program to implement the numerical calculation method for simulating two-dimensional crack propagation according to any one of claims 1-8.

10. A computer-readable storage medium having a computer program stored thereon, characterized in that, When the computer program is executed by the processor, it implements the numerical calculation method for simulating two-dimensional crack propagation according to any one of claims 1-8.

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