CN112836413A - Finite element expansion method for fracture mechanics crack tip singular field calculation - Google Patents
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Abstract
The invention relates to an expanded finite element method for calculating a fracture mechanics crack tip singular field, which comprises the following steps: s1: establishing a geometric model containing cracks, obtaining a Bezier unit, inputting material parameters of Young modulus and Poisson ratio, and applying boundary conditions; s2: obtaining projection points of the control points of the Bezier unit corresponding to the geometric model; s3: based on the projection points, a least square meshless method with a selective interpolation characteristic is adopted as an approximate basis function, and a displacement field is obtained; s4: carrying out re-approximation on the interpolation coefficient in the meshless method by using the approximate basis function; s5: mapping the interpolation coefficient in the re-approximated meshless method to enable the interpolation coefficient to correspond to the control point of the Bezier unit; s6: and approximating the displacement field by using the mapped interpolation coefficient, and solving to obtain the final displacement field. According to the method, the crack tip singular stress field can be efficiently captured without additional degrees of freedom, a good system equation condition number can be obtained, and the geometric accuracy of the model can be maintained.
Description
Technical Field
The invention relates to the field of computer aided engineering, in particular to a finite element expanding method for fracture mechanics crack tip singular field calculation.
Background
Finite element numerical simulation based on linear elastic fracture mechanics is one of effective means for evaluating the fatigue life of a service structure and tracing the fracture and damage reasons of the structure.
However, the standard finite element technology requires that the crack surface and the cell boundary are kept consistent, which brings difficulty in dynamic grid division for the simulation of crack propagation. And because the crack tip has the characteristic of stress singularity, namely the stress is infinite at the crack tip, a unit constructed by taking a polynomial as a base is difficult to approximate a physical field with the singularity, so that the numerical precision is greatly lost. The proposal of the extended finite element method provides a uniform solution to the difficulties. However, in practical application, the finite element propagation method finds that the characteristic functions introduced by the crack tip strengthening basis function are linearly related, so that the finally solved system equation has extremely poor condition number, and the condition number is increased by several times in geometric order along with the increase of the number of model nodes. This limits the applicability of the extended finite element method to complex structures.
Although many scholars have proposed and successfully applied methods that overcome the inherent shortcomings of extended finite elements, most of these methods are based on discrete meshes and are mostly applied to linear elements. However, fracture failure is often caused by cracks initiated and propagated by surface defects, and therefore stress calculations on the surface of the structure are critical. Due to the fact that the geometric accuracy of the discrete grids is lost and the accuracy of the linear units is low, reliable surface stress results cannot be provided.
Therefore, how to provide an accurate and efficient method for expanding finite elements for fracture mechanics crack tip singular field calculation is a technical problem to be solved urgently by those skilled in the art.
Disclosure of Invention
The invention provides an expanded finite element method for calculating a fracture mechanics crack tip singular field, which aims to solve the technical problem.
In order to solve the technical problem, the invention provides an expanded finite element method for calculating a fracture mechanics crack tip singular field, which comprises the following steps:
s1: establishing a geometric model containing cracks, obtaining a Bezier unit, inputting material parameters of Young modulus and Poisson ratio, and applying boundary conditions;
s2: obtaining projection points of the control points of the Bezier unit corresponding to the geometric model by adopting a control point projection method;
s3: based on the projection points, a least square meshless method with a selective interpolation characteristic is adopted as an approximate basis function, and a displacement field is obtained;
s4: re-approximating the interpolation coefficient in the meshless method by using the obtained approximate basis function;
s5: mapping the interpolation coefficient in the re-approximated meshless method to make the interpolation coefficient correspond to the control point of the Bajier unit;
s6: and compounding the base function based on the Bezier spline base function and the approximate base function to obtain a base 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 rational form bezier 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 Bezier spline basis function;
s22: calculating the parameter coordinates of the projection points according to a projection rule, wherein the projection rule is as follows:
ξI+ia sequence of parameter intervals belonging to said cell;
s23: calculating the physical coordinates of the projection points by using the Bezier spline basis function and the control points according to the parameter coordinates of the obtained projection 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; rI，J＝RJ(ξ′I) Is a Bezier spline basis function J at a projection point xi'IThe value of (A) is in the form of a matrix of the above formula
Preferably, in step S3, the step of constructing the approximate basis functions having the selected interpolation characteristics includes:
s31: establishing a support domain of target projection points and a set of projection points in the domain
Wherein i is the number of the target projection point, r is the support domain range, and r is the integral multiple of the number of turns of the target projection point support unit;
s32: the constructed approximate basis function is
Wherein p (x) is [1, p ]1(x),p2(x),…]TIs a vector composed of a polynomial and a characteristic function,
motion matrix, δ, being a least squares methodikAs a function of the switching.
Preferably, in step S4, the obtained approximate basis function is used to re-approximate the interpolation coefficient without grid method, which is expressed as
Corresponding matrix form is
Preferably, in step S5, the re-approximated gridless interpolation coefficient is mapped to correspond to the control point of the bezier unit, and any point on the geometric model can be calculated by the control point and the compounded basis function, where the expression is
Wherein the top dashed line indicates that the control point coefficient is the union set of all points of the support domain of the projection points corresponding to the control points of the Bajier unit; r is a bezier spline basis function.
Preferably, in step S6, based on the composite basis function of the bezier spline basis function and the approximate basis function, the displacement field is approximated by the mapped coefficients corresponding to the control points, and the expression is
Wherein u is the displacement field and u is the displacement field,
is a control coefficient.
Compared with the prior art, the finite element expansion method for calculating the fracture mechanics crack tip singular field provided by the invention has the following advantages:
1. the method is different from an extended finite element method requiring introduction of additional degree of freedom, and can efficiently capture the singular stress field of the crack tip without the additional degree of freedom;
2. the singular function in the invention is contained in the grid-free basis function, and can obtain a good system equation condition number;
3. the accuracy of the geometric model in the invention 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 crack tip singular field calculation in accordance with 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 view of the crack of FIG. 2;
FIG. 4 is a schematic diagram illustrating a range of support points in accordance with one embodiment of the present invention;
FIG. 5 shows the displacement L obtained in one embodiment of the present invention2Comparing the norm error with the schematic diagram;
FIG. 6 is a diagram illustrating a comparison of energy norm errors obtained in an embodiment of the present invention.
In the figure: 1-square area, 2-crack.
Detailed Description
In order to more thoroughly express the technical scheme of the invention, the following specific examples are listed to demonstrate the technical effect; it is emphasized that these examples are intended to illustrate the invention and are not to be construed as limiting the scope of the invention.
The invention provides an extended finite element method for fracture mechanics crack tip singular field calculation, which comprises the following steps as shown in figure 1:
s1: establishing a geometric model containing cracks, obtaining a Bezier unit, inputting material parameters of Young modulus and Poisson ratio, and applying boundary conditions.
Preferably, in step S1, the geometric model is represented by rational form bezier spline, and the accuracy of the geometric figure can be maintained. As shown in fig. 2 and 3, the present embodiment addresses the problem of cracks contained in an infinite flat plate, and selects a square region 1 containing a crack tip with a side length L of 2a, where a is the length of a crack 2 contained in the square region 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 coincident boundaries, being zero plane force boundaries. The first type of boundary condition is imposed around square region 1, and the specific displacement value is given by:
wherein, KI，KIITaking K as a stress intensity factorI＝1，K II0. μ, κ is a material parameter relating young's modulus E to poisson ratio v. For the in-plane strain condition:
κ＝3-4v。
s2: and obtaining projection points of the control points of the Bezier unit corresponding to 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 Bezier spline basis function; in this example, taking p as 2, the parameter interval sequence of the unit is obtained as [0, 0, 0, 1, 1, 1 ].
S22: calculating the parameter coordinates of the projection points according to a projection rule, wherein the projection rule is as follows:
ξI+ia sequence of parameter intervals belonging to said cell;
s23: calculating the physical coordinates of the projection points by using the Bezier spline basis function and the control points according to the parameter coordinates of the obtained projection points, wherein the calculation formula is as follows:
where the left column vector of the equation is the projected point sequence and the right column vectorIs a control point sequence; rI，J＝RJ(ξ′I) Is a Bezier spline basis function J at a projection point xi'IThe value of (A) is in the form of a matrix of the above formula
S3: and based on the projection points, adopting a least square meshless method with a selective interpolation characteristic as an approximate basis function, and obtaining a displacement field.
Preferably, in step S3, the step of constructing the approximate basis functions having the selected interpolation characteristics includes:
s31: establishing a support domain of target projection points and a set of projection points in the domain
Wherein i is the number of the target projection point, r is the range of the support domain, and r is the integral multiple of the number of turns of the target projection point support unit. Fig. 4 shows the case where r is h and r is 2h for the support domain range of the projected point at the cell boundary inside the cell for p is 2.
S32: the constructed approximate basis function is
Wherein p (x) is [1, p ]1(x)，p2(x)，…]TIs a vector composed of a polynomial and a characteristic function,
motion matrix, δ, being a least squares methodikAs a function of the switching.
In this embodiment, a vector is selected:
to reflect the singularity and triangular distribution of the split tip.
S4: and re-approximating the interpolation coefficient in the meshless 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, which is expressed as
Corresponding matrix form is
S5: and mapping the interpolation coefficient in the unapproximated meshless method to enable the interpolation coefficient to correspond to the control point of the Bezier unit.
Preferably, in step S5, the re-approximated gridless interpolation coefficient is mapped to correspond to the control point of the bezier unit, and any point on the geometric model can be calculated by the control point and the compounded basis function, where the expression is
The top dashed lines show that the control point coefficients are increased compared with the control points of the Bezier units and are a union set of all points of the support domain of the projection points corresponding to the control points of the Bezier units; r is a bezier spline basis function.
S6: and compounding the base function based on the Bezier spline base function and the approximate base function to obtain a base 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, based on the composite basis function of the bezier spline basis function and the approximate basis function, the displacement field is approximated by the mapped coefficients corresponding to the control points, and the expression is
Wherein u is the displacement field and u is the displacement field,
is a control coefficient.
FIGS. 5 and 6 show the cell size h of the present embodiment decreasing with mesh subdivision, respectively, and L thereof2The convergence condition of norm and energy norm errors shows that compared with the conventional Bezier spline finite element method, the extended Bezier spline finite element method provided by the invention can improve the result precision by at least two orders of magnitude when being used for calculating the split tip singular field.
Table 1 shows the required iteration number of the iterative solver obtained from all grids when solving the system equation, and the iteration number is positively correlated with the condition number of the system equation, so that the difference of the condition number can be reflected.
TABLE 1
As can be seen from the observation of the table 1, the iteration times of the extended Bezier spline finite element method provided by the invention are similar to those of the conventional finite element method, which shows that the condition number of the corresponding system equation is also similar to that of the conventional finite element under the same mesh and degree of freedom, and the condition number growth rate is also similar along with the mesh subdivision. Therefore, the singular functions in the invention are included in the meshless basis functions, and good system equation condition numbers can be obtained.
In summary, the finite element expansion method for fracture mechanics crack tip singular field calculation provided by the invention comprises the following steps: s1: establishing a geometric model containing cracks, obtaining a Bezier unit, inputting material parameters of Young modulus and Poisson ratio, and applying boundary conditions; s2: obtaining projection points of the control points of the Bezier unit corresponding to the geometric model by adopting a control point projection method; s3: based on the projection points, a least square meshless method with a selective interpolation characteristic is adopted as an approximate basis function, and a displacement field is obtained; s4: re-approximating the interpolation coefficient in the meshless method by using the obtained approximate basis function; s5: mapping the interpolation coefficient in the re-approximated meshless method to make the interpolation coefficient correspond to the control point of the Bajier unit; s6: and compounding the base function based on the Bezier spline base function and the approximate base function to obtain a base 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 adding a degree of freedom; the singular function in the invention is contained in the grid-free basis function, and can obtain a good system equation condition number; the accuracy of the geometric model in the invention can be maintained, and the accuracy of the geometric model is ensured.
It will be apparent to those skilled in the art that various changes and modifications may be made in the invention without departing from the spirit and scope of the invention. Thus, if such modifications and variations of the present invention fall within the scope of the claims of the present invention and their equivalents, the present invention is also intended to include such modifications and variations.
Claims (7)
1. An extended finite element method for fracture mechanics crack tip singular field calculation is characterized by comprising the following steps:
s1: establishing a geometric model containing cracks, obtaining a Bezier unit, inputting material parameters of Young modulus and Poisson ratio, and applying boundary conditions;
s2: obtaining projection points of the control points of the Bezier unit corresponding to the geometric model by adopting a control point projection method;
s3: based on the projection points, a least square meshless method with a selective interpolation characteristic is adopted as an approximate basis function, and a displacement field is obtained;
s4: re-approximating the interpolation coefficient in the meshless method by using the obtained approximate basis function;
s5: mapping the interpolation coefficient in the re-approximated meshless method to make the interpolation coefficient correspond to the control point of the Bajier unit;
s6: and compounding the base function based on the Bezier spline base function and the approximate base function to obtain a base 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 calculation of claim 1, wherein in step S1, the geometric model is represented using rational form bezier splines.
3. The extended finite element method for singular field calculation of a fracture mechanical crack tip as claimed in claim 1, wherein in step S2, the step of obtaining the projected points by 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 Bezier spline basis function;
s22: calculating the parameter coordinates of the projection points according to a projection rule, wherein the projection rule is as follows:
ξI+ia sequence of parameter intervals belonging to said cell;
s23: calculating the physical coordinates of the projection points by using the Bezier spline basis function and the control points according to the parameter coordinates of the obtained projection 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; rI，J＝RJ(ξ′I) Is a Bezier spline basis function J at a projection point xi'IThe value of (A) is in the form of a matrix of the above formula
4. The extended finite element method for singular field calculation of a fracture mechanics crack tip of claim 1, wherein in step S3, the step of constructing the approximated basis functions with selected interpolation characteristics comprises:
s31: establishing a support domain of target projection points and a set of projection points in the domain
Wherein i is the number of the target projection point, r is the support domain range, and r is the integral multiple of the number of turns of the target projection point support unit;
s32: the constructed approximate basis function is
Wherein p (x) is [1, p ]1(x)，p2(x)，…]TIs a vector composed of a polynomial and a characteristic function,
motion matrix, δ, being a least squares methodikAs a function of the switching.
5. The extended finite element method for singular field computation of fracture mechanics crack tip as claimed in claim 1, wherein in step S4, said obtained approximate basis functions are used to re-approximate the interpolation coefficients without grid method, wherein the expression is
Corresponding matrix form is
6. The extended finite element method of claim 1, wherein in step S5, the re-approximated meshless interpolation coefficients are mapped to control points of the bezier unit, and any point on the geometric model can be calculated from the control points and the compounded basis function, and the expression is
Wherein the top dashed line indicates that the control point coefficient is the union set of all points of the support domain of the projection points corresponding to the control points of the Bajier unit; r is a bezier spline basis function.
7. The extended finite element method for fracture mechanics split tip singular field calculation of claim 1, wherein in step S6, based on the base function of the composite of the bezier spline base function and the approximate base function, the displacement field is approximated by the mapped coefficients corresponding to the control points, which is expressed as
Wherein u is the displacement field and u is the displacement field,
is a control coefficient.
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