CN112836413B - Extended finite element method for calculating singular field of fracture mechanics fracture tip - Google Patents
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Abstract
The invention relates to an extended finite element method for calculating a fracture mechanics fracture tip singular field, which comprises the following steps: s1: establishing a geometric model containing cracks, obtaining Bei Jier units, inputting material parameters of Young modulus and Poisson ratio, and applying boundary conditions; s2: obtaining a projection point of a control point of the Bei Jier unit on the geometric model; s3: a least square non-grid method with interpolation selection characteristic is adopted as an approximate basis function based on the projection points, and a displacement field is obtained; s4: re-approximating the interpolation coefficient in the gridless method by using the approximate basis function; s5: mapping interpolation coefficients in the re-approximated gridless method to enable the interpolation coefficients to correspond to control points of Bei Jier units; s6: and approximating the displacement field by using the mapped interpolation coefficient, and solving to obtain a final displacement field. The method can efficiently capture the singular stress field of the fracture tip without additional degrees of freedom, can obtain good system equation condition number, and can maintain the geometric accuracy of the model.
Description
Technical Field
The invention relates to the field of computer aided engineering, in particular to an extended finite element method for calculating a singular field of a fracture mechanics fracture tip.
Background
Finite element numerical simulation based on linear elastic fracture mechanics is one of the effective means for evaluating fatigue life of a service structure and tracing structural fracture and damage reasons.
However, standard finite element techniques require that the crack plane be consistent with the cell boundary, which presents difficulties in grid dynamic partitioning for simulation of crack propagation. And because the crack tip has the characteristic of singular stress, namely the stress is infinitely large at the crack tip, the unit with the polynomial as the base is difficult to approximate to the physical field with the singular characteristic, thereby greatly losing the numerical precision. The proposal of the extended finite element method provides a unified solution to the above difficulties. However, the extended finite element method finds that the feature function introduced by the crack tip strengthening basis function is linearly related in practical application, so that the finally solved system equation has extremely poor condition number, and the condition number is increased in geometric series multiple with the increase of the number of model nodes. This limits the application of the extended finite element method in complex structures.
Although many scholars have proposed and successfully applied methods to overcome the inherent disadvantages of extended finite elements, most of these methods are based on discrete grids and are applied to linear cells. Fracture failure, however, is often caused by the initiation and propagation of cracks in surface defects, and therefore stress calculations on the surface of the structure are critical. The discrete grid often fails to give reliable surface stress results due to the loss of geometric accuracy and the lower accuracy of the linear cells.
Therefore, how to provide an accurate and efficient extended finite element method for calculating the singular field of fracture mechanics fracture tip is a technical problem to be solved by those skilled in the art.
Disclosure of Invention
The invention provides an extended finite element method for calculating a singular field of a fracture mechanics fracture tip, which aims to solve the technical problems.
In order to solve the technical problems, the invention provides an extended finite element method for calculating a singular field of a fracture mechanics fracture tip, which comprises the following steps:
s1: establishing a geometric model containing cracks, obtaining Bei Jier units, inputting material parameters of Young modulus and Poisson ratio, and applying boundary conditions;
s2: obtaining a projection point of a control point of the Bei Jier unit on the geometric model by adopting a control point projection method;
s3: a least square non-grid method with interpolation selection characteristic is adopted as an approximate basis function based on the projection points, and a displacement field is obtained;
s4: re-approximating the interpolation coefficient in the gridless method by using the obtained approximate basis function;
s5: mapping the interpolation coefficient in the re-approximated gridless method to correspond to the control point of the Bei Jier unit;
s6: and (3) based on the Bei Jier spline basis function and the approximate basis function, obtaining a basis function, approximating the displacement field by using the mapped interpolation coefficient corresponding to the control point, calculating a finite element stiffness equation and solving to obtain a final displacement field.
Preferably, in step S1, the geometric model is represented by a rational form Bei Jier spline.
Preferably, in step S2, the step of obtaining the projection point by using the control point projection method includes:
s21: setting the repetition degree of head and tail elements of a unit parameter interval [0,1] as p+1 to obtain a parameter interval sequence of a unit, wherein p is the order of the Bei Jier spline basis function;
s22: calculating parameter coordinates of the projection points according to a projection rule, wherein the projection rule is as follows:ξ I+i a sequence of parameter intervals belonging to said unit;
s23: according to the parameter coordinates of the obtained projection points, calculating physical coordinates of the projection points by using the Bei Jier spline basis function and the control points, wherein the calculation formula is as follows:
wherein equation isThe left column vector is a projection point sequence, and the right column vector is a control point sequence; r is R I，J ＝R J (ξ′ I ) For Bei Jier spline basis function J at projection point ζ' I The values of the above formula are in the form of a matrix
Preferably, in step S3, the step of constructing the approximate basis function having the selected interpolation characteristic includes:
s31: establishing a supporting domain of target projection points and a set of projection points in the domainWherein i is the number of the target projection point, r is the supporting domain range, and r is the integer multiple of the number of turns of the supporting unit of the target projection point;
s32: the constructed approximate basis function is
Wherein p (x) = [1, p 1 (x),p 2 (x),…] T Is a vector composed of polynomials and feature functions,motion matrix delta as least square method ik As a switching function.
Preferably, in step S4, the obtained approximate basis function is used to re-approximate the interpolation coefficient without grid method, where the expression isThe corresponding matrix form is->
Preferably, in step S5, the re-approximated interpolation coefficient without grid method is mapped to correspond to the control point of Bei Jier unit, any point on the geometric model can be calculated by the control point and the combined basis function, and the expression is as follows
Wherein the top dashed line indicates that the control point coefficient is the union of all points of the support domain of the projection point corresponding to the Bei Jier unit control point; r is Bei Jier spline basis function.
Preferably, in step S6, the displacement field is approximated by using the mapped coefficient corresponding to the control point based on the basis function obtained by compositing the Bei Jier spline basis function and the approximate basis function, where the expression is thatWherein u is the displacement field, ">Is the control coefficient.
Compared with the prior art, the extended finite element method for calculating the singular field of the fracture mechanics fracture tip has the following advantages:
1. the method is different from an extended finite element method requiring the introduction of additional degrees of freedom, and the method can efficiently capture the singular stress field of the crack tip without the additional degrees of freedom;
2. the singular function in the invention is contained in the gridless basis function, so that good system equation condition number can be obtained;
3. the accuracy of the geometric model can be maintained, and the accuracy of the geometric model is ensured.
Drawings
FIG. 1 is a flow chart of an extended finite element method for fracture mechanics fracture tip singular field calculation in an embodiment of the present invention;
FIG. 2 is a schematic diagram of a geometric model according to an embodiment of the present invention;
FIG. 3 is a schematic illustration of the crack of FIG. 2;
FIG. 4 is a schematic diagram illustrating the range of support points according to an embodiment of the present invention;
FIG. 5 shows the resulting displacement L in an embodiment of the present invention 2 A norm error comparison diagram;
FIG. 6 is a graph showing energy norm error versus time for an embodiment of the present invention.
In the figure: 1-square area, 2-crack.
Detailed Description
In order to describe the technical solution of the above invention in more detail, the following specific examples are listed to demonstrate technical effects; it is emphasized that these examples are illustrative of the invention and are not limiting the scope of the invention.
The invention provides an extended finite element method for calculating a fracture mechanics fracture tip singular field, which is shown in figure 1 and comprises the following steps:
s1: and establishing a geometric model containing cracks, obtaining Bei Jier units, inputting material parameters of Young's modulus and Poisson's ratio, and applying boundary conditions.
Preferably, in step S1, the geometric model is represented by a spline of rational form Bei Jier, whose geometric accuracy can be maintained. As shown in fig. 2 and 3, the object of the present embodiment is to solve the problem of cracks contained in an infinite flat plate, and to select a square area 1 having a side length l=2a of the tip of the crack, where a is the length of the crack 2 contained in the square area 1. The crack 2 is located in the center of the square area 1.
With continued reference to fig. 3, the crack 2 is formed by two overlapping boundaries, which are zero-plane force boundaries. The boundary conditions of the first type are applied around the square area 1, and the specific displacement value is given by the following formula:
wherein K is I ，K II Taking K as stress intensity factor I ＝1，K II =0. μ, κ is a material parameter related to poisson's ratio v at young's modulus E. For planar strain conditions:κ＝3-4v。
s2: and obtaining the projection points of the control points of the Bei Jier unit on the geometric model by adopting a control point projection method.
Preferably, in step S2, the step of obtaining the projection point by using the control point projection method includes:
s21: setting the repetition degree of head and tail elements of a unit parameter interval [0,1] as p+1 to obtain a parameter interval sequence of a unit, wherein p is the order of the Bei Jier spline basis function; in this example, taking p=2, the parameter interval sequence of the cell is [0,0,0,1,1,1].
S22: calculating parameter coordinates of the projection points according to a projection rule, wherein the projection rule is as follows:ξ I+i a sequence of parameter intervals belonging to said unit;
s23: according to the parameter coordinates of the obtained projection points, calculating physical coordinates of the projection points by using the Bei Jier spline basis function and the control points, wherein the calculation formula is as follows:
wherein the left column vector of the equation is a projection point sequence, and the right column vector is a control point sequence; r is R I，J ＝R J (ξ′ I ) For Bei Jier spline basis function J at projection point ζ' I The values of the above formula are in the form of a matrix
S3: and adopting a least square non-grid method with selective interpolation characteristic as an approximate basis function based on the projection points, and obtaining a displacement field.
Preferably, in step S3, the step of constructing the approximate basis function having the selected interpolation characteristic includes:
s31: establishing a supporting domain of target projection points and a set of projection points in the domainWherein i is the number of the target projection point, r is the supporting domain range, and r is the integer multiple of the number of turns of the supporting unit of the target projection point. Fig. 4 shows the case where r=h and r=2h are taken for the support domain range of the projected points inside the cell and at the cell boundary in one dimension for p=2.
S32: the constructed approximate basis function is
Wherein p (x) = [1, p 1 (x)，p 2 (x)，…] T Is a vector composed of polynomials and feature functions,motion matrix delta as least square method ik As a switching function.
In this embodiment, the vector is selected:
to reflect the split tip singularity and triangular distribution.
S4: and re-approximating the interpolation coefficient in the gridless method by using the obtained approximate basis function.
Preferably, in step S4, the obtained approximate basis function is used to re-approximate the interpolation coefficient without grid method, where the expression isThe corresponding matrix form is->
S5: the interpolation coefficient in the re-approximated gridless method is mapped so as to correspond to the control point of the Bei Jier unit.
Preferably, in step S5, the re-approximated interpolation coefficient without grid method is mapped to correspond to the control point of Bei Jier unit, any point on the geometric model can be calculated by the control point and the combined basis function, and the expression is as follows
Wherein the top dashed line indicates that the control point coefficient is increased compared with the Bei Jier unit control point, and is the union of all points of the support domain of the projection point corresponding to the Bei Jier unit control point; r is Bei Jier spline basis function.
S6: and (3) based on the Bei Jier spline basis function and the approximate basis function, obtaining a basis function, approximating the displacement field by using the mapped interpolation coefficient corresponding to the control point, calculating a finite element stiffness equation and solving to obtain a final displacement field.
Preferably, in step S6, the displacement field is approximated by using the mapped coefficient corresponding to the control point based on the basis function obtained by compositing the Bei Jier spline basis function and the approximate basis function, where the expression is thatWherein u is the displacement field, ">Is the control coefficient.
Fig. 5 and 6 show respectivelyThe embodiment is shown with decreasing cell size h as the grid subdivides, L 2 The convergence of the norm and energy norm errors shows that compared with the conventional Bei Jier spline finite element method, the extended Bei Jier spline finite element method provided by the invention can improve the result accuracy by at least two orders of magnitude when being used for calculating the crack tip singular field.
Table 1 shows the number of iterations required by the resulting iterative solver for all meshes in solving the system equation, which is positively correlated with the condition number of the system equation, and thus can reflect the difference in condition number.
TABLE 1
As can be seen from table 1, the number of iterations of the extended Bei Jier spline finite element method provided by the present invention is similar to that of the conventional finite element method, which indicates that the condition number of the corresponding system equation is similar to that of the conventional finite element under the same grid and degree of freedom, and the condition number growth rate is similar as the grid is subdivided. Thus, the singular function of the present invention is included in the grid-free basis function, and a good condition number of the system equation can be obtained.
In summary, the extended finite element method for calculating the singular field of the fracture mechanics fracture tip provided by the invention comprises the following steps: s1: establishing a geometric model containing cracks, obtaining Bei Jier units, inputting material parameters of Young modulus and Poisson ratio, and applying boundary conditions; s2: obtaining a projection point of a control point of the Bei Jier unit on the geometric model by adopting a control point projection method; s3: a least square non-grid method with interpolation selection characteristic is adopted as an approximate basis function based on the projection points, and a displacement field is obtained; s4: re-approximating the interpolation coefficient in the gridless method by using the obtained approximate basis function; s5: mapping the interpolation coefficient in the re-approximated gridless method to correspond to the control point of the Bei Jier unit; s6: and (3) based on the Bei Jier spline basis function and the approximate basis function, obtaining a basis function, approximating the displacement field by using the mapped interpolation coefficient corresponding to the control point, calculating a finite element stiffness equation and solving to obtain a final displacement field. The method can efficiently capture the singular stress field of the crack tip without additional degrees of freedom; the singular function in the invention is contained in the gridless basis function, so that good system equation condition number can be obtained; the accuracy of the geometric model can be maintained, and the accuracy of the geometric model is ensured.
It will be apparent to those skilled in the art that various modifications and variations can be made to the present invention without departing from the spirit or scope of the invention. Thus, it is intended that the present invention also include such modifications and alterations insofar as they come within the scope of the appended claims or the equivalents thereof.
Claims (7)
1. An extended finite element method for fracture mechanics fracture tip singular field calculation, comprising the steps of:
s1: establishing a geometric model containing cracks, obtaining Bei Jier units, inputting material parameters of Young modulus and Poisson ratio, and applying boundary conditions;
s2: obtaining a projection point of a control point of the Bei Jier unit on the geometric model by adopting a control point projection method;
s3: a least square non-grid method with interpolation selection characteristic is adopted as an approximate basis function based on the projection points, and a displacement field is obtained;
s4: re-approximating the interpolation coefficient in the gridless method by using the obtained approximate basis function;
s5: mapping the interpolation coefficient in the re-approximated gridless method to correspond to the control point of the Bei Jier unit;
s6: and (3) based on the Bei Jier spline basis function and the approximate basis function, obtaining a basis function, approximating the displacement field by using the mapped interpolation coefficient corresponding to the control point, calculating a finite element stiffness equation and solving to obtain a final displacement field.
2. The extended finite element method for fracture mechanics split tip singular field computation of claim 1, wherein in step S1, the geometric model is represented in a rational form Bei Jier spline.
3. The extended finite element method for fracture mechanics split tip singular field calculation of claim 1, wherein in step S2, the step of obtaining the proxels using the control point projection method comprises:
s21: setting the repetition degree of head and tail elements of a unit parameter interval [0,1] as p+1 to obtain a parameter interval sequence of a unit, wherein p is the order of the Bei Jier spline basis function;
s22: calculating parameter coordinates of the projection points according to a projection rule, wherein the projection rule is as follows:ξ I+i a sequence of parameter intervals belonging to said unit;
s23: according to the parameter coordinates of the obtained projection points, calculating physical coordinates of the projection points by using the Bei Jier spline basis function and the control points, wherein the calculation formula is as follows:
wherein the left column vector of the equation is a projection point sequence, and the right column vector is a control point sequence; r is R I，J ＝R J (ξ′ I ) For Bei Jier spline basis function J at projection point ζ' I The values of the above formula are in the form of a matrix
4. The extended finite element method for fracture mechanics split tip singular field calculation of claim 1, wherein in step S3, the step of constructing the approximate basis function with selected interpolation characteristics comprises:
s31: establishing a supporting domain of target projection points and a set of projection points in the domainWherein i is the number of the target projection point, r is the supporting domain range, and r is the integer multiple of the number of turns of the supporting unit of the target projection point;
s32: the constructed approximate basis function is
Wherein p (x) = [1, p 1 (x)，p 2 (x)，…] T Is a vector composed of polynomials and feature functions,motion matrix delta as least square method ik As a switching function.
5. The extended finite element method for fracture mechanics split tip singular field calculation according to claim 1, wherein in step S4, the obtained approximate basis function is used to re-approximate the grid-free interpolation coefficients, expressed asThe corresponding matrix form is->
6. The extended finite element method for fracture mechanics split tip singular field calculation according to claim 1, wherein in step S5, the re-approximated meshfree interpolation coefficient is mapped to correspond to the control point of the Bei Jier unit, any point on the geometric model is calculated from the control point and the compounded basis function, and the expression is
Wherein the top dashed line indicates that the control point coefficient is the union of all points of the support domain of the projection point corresponding to the Bei Jier unit control point; r is Bei Jier spline basis function.
7. The extended finite element method for fracture mechanics fracture tip singular field calculation according to claim 1, wherein in step S6, the displacement field is approximated by using the mapped coefficient corresponding to the control point based on the basis function of the Bei Jier spline basis function and the approximate basis function, which is expressed asWhere u is the displacement field and where,is the control coefficient.
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