Method for enhancing structural nonlinearity solution convergence by adjusting node displacement

Abstract

The invention discloses a method for enhancing structural nonlinear solution convergence by adjusting node displacement. The method comprises the following steps: S1, establishing a geometric model; s2, acquiring grid information; s3, body loads, surface loads and line loads are converted into node equivalent loads according to the virtual work principle; s4, setting a main control node, enabling the main control node not to belong to the node of the grid model, and selecting a node to which a load should be applied from the grid model; constructing a constraint equation; s5, forming an overall stiffness matrix, and then introducing the constraint equation into the overall stiffness matrix to obtain a nonlinear equation set; and S6, solving the nonlinear equation set to obtain the node displacement of the to-be-analyzed structure in each increment step, and obtaining the displacement of the node to which the load should be applied based on the node displacement of the to-be-analyzed structure in each increment step. According to the method, the problem that solution cannot be achieved due to singularity of the tangent stiffness matrix is solved, the convergence of nonlinear solution of the structure is enhanced, and meanwhile the problem that the iterative process is complex in arc length method calculation is avoided.

Description

Technical Field

[0001] The invention relates to the technical field of structural mechanics simulation of computer-aided engineering, and in particular to a method for enhancing the convergence of structural nonlinear solutions by adjusting node displacements. Background Art

[0002] With the rapid development of current industry, structural simulation models tend to be increasingly complex. In actual simulations, convergence problems may arise due to complex models, complex geometry, and complex working conditions.

[0003] Commonly used methods in structural nonlinear analysis include: (1) Newton-Raphson iteration method. The Newton-Raphson iteration method is used to solve the nonlinear equation group. In fact, it is to linearize the nonlinear equation group into multiple linear equation groups by linearization to achieve nonlinear solution. In each step, the displacement increment is solved, and the current stress and the current tangent stiffness matrix are obtained based on the displacement increment. The current tangent stiffness matrix is ​​used to continue to solve the next increment. In actual analysis, there may be a variety of situations that lead to the singularity of the tangent stiffness matrix, such as structural instability or material fracture. At this time, due to the singularity of the tangent stiffness matrix, the Newton-Raphson iteration method cannot solve the next incremental displacement; (2) Load increment solution method, which is to convert the load to be applied into the load increment. The load is gradually loaded in multiple sub-increments. After solving each sub-increment, the incremental displacements of the sub-increment are superimposed. At the end of each incremental step, the tangent stiffness matrix is ​​calculated for the calculation of the incremental displacement of the next incremental step. However, since the tangent stiffness matrix may be singular, the accumulated deviation in each incremental step may cause the calculation result to deviate from the true solution. (3) The arc length method is to add the unknown number of the load coefficient to the original nonlinear equation group. During the solution process, the load coefficient and the increment are controlled at the same time. The constraint equations corresponding to the load coefficient and displacement increment are selected as the control equations to calculate the final result. However, the calculation iteration process of this method is relatively complicated. For different problems, it is necessary to select appropriate constraint control equations based on experience to achieve rapid convergence. Summary of the invention

[0004] The present invention provides a method for enhancing the convergence of structural nonlinear solutions by adjusting node displacements, so as to overcome the technical problems in existing methods adopted for structural nonlinear analysis, such as failure to converge structural nonlinear solutions or complex calculation iteration processes due to various limitations.

[0005] In order to achieve the above object, the technical solution of the present invention is:

[0006] A method for enhancing the convergence of a structural nonlinear solution by adjusting node displacements, the specific steps comprising:

[0007] S1: Establishing a geometric model of the structure to be analyzed, meshing the geometric model to obtain a mesh model for finite element calculation, setting boundary conditions for the mesh model, and obtaining the total number of nodes, total number of elements, and total number of degrees of freedom of the mesh model;

[0008] S2: Obtaining grid information, wherein the grid information includes grid node information, unit information, boundary information, and the topological relationship between the three;

[0009] S3: Obtaining load information, the load information includes body load, surface load, line load and point load, and converting the body load, surface load and line load into node equivalent load using the principle of virtual work;

[0010] S4: Setting a main control node, and making the main control node not a node of the grid model, and selecting a node to which the load should be applied from the grid model; using the virtual work principle and based on the node equivalent load and the point load, obtaining the node displacement coefficient at the node to which the load should be applied and the node displacement coefficient at the main control node, thereby constructing a constraint equation;

[0011] S5: obtaining a unit stiffness matrix of the mesh model according to the unit information, and forming an overall stiffness matrix based on the unit stiffness matrix, and then introducing a constraint equation into the overall stiffness matrix, thereby obtaining a nonlinear equation group formed by the overall stiffness matrix considering the constraint equation;

[0012] S6: The Newton-Raphson method is used to solve the nonlinear equation group. After the solution converges, the node displacements of the structure to be analyzed in each incremental step are obtained, and the displacements of the nodes to which the load should be applied are obtained based on the node displacements of the structure to be analyzed in each incremental step.

[0013] Furthermore, in S3, the formula for converting body load, surface load and line load into node equivalent load using the principle of virtual work is:

[0014]

[0015] In the formula, R i is the equivalent nodal load, N i is the unit shape function, F v is the body load distribution, F s is the surface load distribution, F l is the linear load distribution, and Ω is the corresponding distributed load integration area.

[0016] Furthermore, in S4, the constructed constraint equation is expressed as:

[0017]

[0018] In the formula, F iis the node displacement coefficient at the node where the load should be applied, that is, the load to be applied for the i-th degree of freedom, including node equivalent load and point load; u i is the displacement corresponding to the ith degree of freedom; u n+1 is the displacement corresponding to the main control node; is the nodal displacement coefficient at the main control node; n is the total number of nodes to which the load should be applied.

[0019] Further, in S5, the process of introducing the constraint equation into the overall stiffness matrix is:

[0020] S51: Obtain the degree of freedom of the main control node corresponding to the constraint equation and the degree of freedom of each node to which the load should be applied;

[0021] S52: Get the maximum value K of the elements in the global stiffness matrix MAX , and define the correction coefficient P, P = 1e5*K MAX ;

[0022] S53: Modify the diagonal and off-diagonal elements of the degrees of freedom corresponding to the constraint equations in the global stiffness matrix, including:

[0023] a) The diagonal element K of the degrees of freedom of each node to which the load should be applied ii Add the first correction value, the first correction value is P*F i *F i ;

[0024] b) The diagonal elements K of the degrees of freedom of the main control nodes jj Add a second correction value, the second correction value is

[0025] c) Add a third correction value to the off-diagonal elements formed by the row corresponding to the degree of freedom of the main control node and the column j corresponding to the degree of freedom of each node to which the load should be applied. The third correction value is

[0026] d) Add the third correction value to the off-diagonal elements formed by the column corresponding to the degree of freedom of the main control node and the row j corresponding to the degree of freedom of each node to which the load should be applied, that is,

[0027] Further, in S6, the Newton-Raphson method is used to solve the nonlinear equation group, and after the solution converges, the node displacements of the structure to be analyzed at each incremental step are obtained, and the process of obtaining the displacements of the nodes to which the load should be applied based on the node displacements of the structure to be analyzed at each incremental step is as follows:

[0028] S61: using the Newton-Raphson method to solve the nonlinear equations, and after the solution converges, obtaining the node displacements of the structure to be analyzed at each incremental step;

[0029] S62: calculating the constraint reaction force of the structure to be analyzed at the boundary based on the node displacement of the structure to be analyzed in each incremental step, and obtaining a constraint reaction force variation curve;

[0030] S63: Compare the magnitude of each value in the constraint reaction force change curve with the sum of the original load values ​​that should be applied in turn. If the displacement value of the current incremental step in the constraint reaction force change curve is less than the sum of the original load values ​​that should be applied, and the displacement value in the next incremental step is greater than the sum of the original load values ​​that should be applied, it means that the displacement value of the structure to be analyzed under the original load that should be applied corresponds to the current incremental step. Then the current incremental step is the incremental step corresponding to the original load value that should be applied. Interpolation is performed based on the displacement between the beginning and the end of the current incremental step, that is, the displacement corresponding to the node of the original load value that should be applied is obtained.

[0031] Beneficial effect: The present invention establishes a geometric model of the structure to be analyzed and obtains a mesh model for finite element calculation, sets a main control node and selects several nodes to which loads should be applied, and establishes constraint equations based on the main control node and several nodes to which loads should be applied, and then introduces the constraint equations into the overall stiffness matrix, so as to achieve displacement only on the main control node, so that under the condition of no load being applied, the proportion of the load that should be applied is completely consistent, thereby meeting the applied load requirement. On the basis of this condition, the Newton-Raphson iterative method can be used to solve the displacement change of each node, which solves the problem that the tangent stiffness matrix singularity leads to the inability to solve the problem, enhances the convergence of the structural nonlinear solution, and avoids the problem of complex iterative process of arc length method calculation. BRIEF DESCRIPTION OF THE DRAWINGS

[0032] In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the following briefly introduces the drawings required for use in the embodiments or the description of the prior art. Obviously, the drawings described below are some embodiments of the present invention. For ordinary technicians in this field, other drawings can be obtained based on these drawings without paying creative labor.

[0033] Figure 1 It is a flow chart of the method for enhancing the convergence of structural nonlinear solution by adjusting node displacement in the present invention;

[0034] Figure 2 It is a front view of the lattice shell structure in an embodiment of the present invention;

[0035] Figure 3 A top view of a lattice shell structure in an embodiment of the present invention;

[0036] Figure 4 is a distribution diagram of load nodes in an embodiment of the present invention;

[0037] Figure 5 The figure is a comparison diagram of the load-displacement curves calculated by the present method and the arc length method in the embodiment of the present invention. DETAILED DESCRIPTION

[0038] In order to make the purpose, technical solution and advantages of the embodiments of the present invention clearer, the technical solution in the embodiments of the present invention will be clearly and completely described below in conjunction with the drawings in the embodiments of the present invention. Obviously, the described embodiments are part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without creative work are within the scope of protection of the present invention.

[0039] This embodiment provides a method for enhancing the convergence of structural nonlinear solutions by adjusting node displacements, such as Figure 1 As shown, a method for enhancing the convergence of structural nonlinear solutions by adjusting node displacements is characterized in that the specific steps include:

[0040] S1: Establishing a geometric model of the structure to be analyzed, meshing the geometric model to obtain a mesh model for finite element calculation, setting boundary conditions for the mesh model, and obtaining the total number of nodes, total number of elements, and total number of degrees of freedom of the mesh model;

[0041] S2: Obtaining grid information, wherein the grid information includes grid node information, unit information, boundary information, and the topological relationship between the three;

[0042] S3: Obtaining load information, the load information includes body load, surface load, line load and point load, and converting the body load, surface load and line load into node equivalent load using the principle of virtual work;

[0043] In a specific embodiment, in S3, the formula for converting body load, surface load and line load into node equivalent load using the virtual work principle is:

[0044]

[0045] In the formula, R i is the equivalent nodal load, N i is the unit shape function, F v is the body load distribution, F s is the surface load distribution, F l is the linear load distribution, and Ω is the corresponding distributed load integration area.

[0046] S4: Setting a main control node, and making the main control node not a node of the grid model, and selecting a node to which the load should be applied from the grid model; using the virtual work principle and based on the node equivalent load and the point load, obtaining the node displacement coefficient at the node to which the load should be applied and the node displacement coefficient at the main control node, thereby constructing a constraint equation;

[0047] In a specific embodiment, in S4, the constructed constraint equation is expressed as:

[0048]

[0049] In the formula, F i is the node displacement coefficient at the node where the load should be applied, that is, the load to be applied for the i-th degree of freedom, including node equivalent load and point load; u i is the displacement corresponding to the ith degree of freedom; u n+1 is the displacement corresponding to the main control node; is the nodal displacement coefficient at the main control node; n is the total number of nodes to which the load should be applied.

[0050] S5: obtaining a unit stiffness matrix of the mesh model according to the unit information, and forming an overall stiffness matrix based on the unit stiffness matrix, and then introducing a constraint equation into the overall stiffness matrix, thereby obtaining a nonlinear equation group formed by the overall stiffness matrix considering the constraint equation;

[0051] In a specific embodiment, in S5, the process of introducing the constraint equation into the overall stiffness matrix is:

[0052] S51: Obtain the degree of freedom of the main control node corresponding to the constraint equation and the degree of freedom of each node to which the load should be applied;

[0053] S52: Get the maximum value K of the elements in the global stiffness matrix MAX , and define the correction coefficient P, P = 1e5*K MAX ;

[0054] S53: Modify the diagonal and off-diagonal elements of the degrees of freedom corresponding to the constraint equations in the global stiffness matrix, including:

[0055] a) The diagonal element K of the degrees of freedom of each node to which the load should be applied ii Add the first correction value, the first correction value is P*F i *F i ;

[0056] b) The diagonal elements K of the degrees of freedom of the main control nodes jj Add a second correction value, the second correction value is

[0057] c) Add a third correction value to the off-diagonal elements formed by the row corresponding to the degree of freedom of the main control node and the column j corresponding to the degree of freedom of each node to which the load should be applied. The third correction value is

[0058] d) Add the third correction value to the off-diagonal elements formed by the column corresponding to the degree of freedom of the main control node and the row j corresponding to the degree of freedom of each node to which the load should be applied, that is,

[0059] S6: The Newton-Raphson method is used to solve the nonlinear equation group. After the solution converges, the node displacements of the structure to be analyzed in each incremental step are obtained, and the displacements of the nodes to which the load should be applied are obtained based on the node displacements of the structure to be analyzed in each incremental step.

[0060] In a specific embodiment, in S6, the Newton-Raphson method is used to solve the nonlinear equation group, and after the solution converges, the node displacements of the structure to be analyzed at each incremental step are obtained, and the displacements of the nodes to which the load should be applied are obtained based on the node displacements of the structure to be analyzed at each incremental step. The process is:

[0061] S61: using the Newton-Raphson method to solve the nonlinear equations, and after the solution converges, obtaining the node displacements of the structure to be analyzed at each incremental step;

[0062] S62: calculating the constraint reaction force of the structure to be analyzed at the boundary based on the node displacement of the structure to be analyzed in each incremental step, and obtaining a constraint reaction force variation curve;

[0063] S63: Compare the magnitude of each value in the constraint reaction force change curve with the sum of the original load values ​​that should be applied in turn. If the displacement value of the current incremental step in the constraint reaction force change curve is less than the sum of the original load values ​​that should be applied, and the displacement value in the next incremental step is greater than the sum of the original load values ​​that should be applied, it means that the displacement value of the structure to be analyzed under the original load that should be applied corresponds to the current incremental step. Then the current incremental step is the incremental step corresponding to the original load value that should be applied. Interpolation is performed based on the displacement between the beginning and the end of the current incremental step, that is, the displacement corresponding to the node of the original load value that should be applied is obtained.

[0064] Embodiment 1:

[0065] like Figure 2 and Figure 3As shown in the figure, a structural nonlinear analysis of a Kvit 6 single-layer lattice shell structure with a span of 32m and a rise of 4.29m is performed. The cross-sectional size of the lattice shell is 80×8mm, the material is an ideal elastic-plastic material, the elastic modulus E=2.0e11N / m2, the Poisson's ratio is 0.3, and the yield strength of the elastic-plastic material is 235Mpa. The bottom of the lattice shell is hinged, 7 nodes are selected from the lattice shell structure, and vertical loads are applied to the 7 nodes of the lattice shell, where the node load values ​​are F z1 =-400000N, F z2 =-30000N, F z3 =-200000N, F z4 =-100000N, the distribution of specific load points is as follows Figure 4 As shown; Based on this model, the method proposed in this embodiment and the ordinary arc length method are used to solve the problem, and the load-displacement curve of the structure to be analyzed under the load is obtained. The calculation results are shown in Figure 5 As shown in the figure, it can be seen that the method proposed in this embodiment can effectively obtain the descending section of the load-displacement curve and has good convergence. At the same time, practice shows that the method proposed in this embodiment has less calculation amount and higher efficiency than the arc length method, and is more conducive to enhancing the convergence of structural nonlinear solutions.

[0066] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention, rather than to limit it. Although the present invention has been described in detail with reference to the aforementioned embodiments, those skilled in the art should understand that they can still modify the technical solutions described in the aforementioned embodiments, or replace some or all of the technical features therein with equivalents. However, these modifications or replacements do not cause the essence of the corresponding technical solutions to deviate from the scope of the technical solutions of the embodiments of the present invention.

Claims

1. A method for enhancing the convergence of structural nonlinear solutions by adjusting node displacements, characterized in that: The specific steps include: S1: Establishing a geometric model of the structure to be analyzed, meshing the geometric model to obtain a mesh model for finite element calculation, setting boundary conditions for the mesh model, and obtaining the total number of nodes, total number of elements, and total number of degrees of freedom of the mesh model; S2: Obtaining grid information, wherein the grid information includes grid node information, unit information, boundary information, and the topological relationship between the three; S3: Obtaining load information, the load information includes body load, surface load, line load and point load, and converting the body load, surface load and line load into node equivalent load using the principle of virtual work; S4: Setting a main control node, and making the main control node not a node of the grid model, and selecting a node to which the load should be applied from the grid model; using the virtual work principle and based on the node equivalent load and the point load, obtaining the node displacement coefficient at the node to which the load should be applied and the node displacement coefficient at the main control node, thereby constructing a constraint equation; S5: obtaining a unit stiffness matrix of the mesh model according to the unit information, and forming an overall stiffness matrix based on the unit stiffness matrix, and then introducing a constraint equation into the overall stiffness matrix, thereby obtaining a nonlinear equation group formed by the overall stiffness matrix considering the constraint equation; S6: The Newton-Raphson method is used to solve the nonlinear equation group. After the solution converges, the node displacements of the structure to be analyzed in each incremental step are obtained, and the displacements of the nodes to which the load should be applied are obtained based on the node displacements of the structure to be analyzed in each incremental step.

2. The method for enhancing the convergence of structural nonlinear solutions by adjusting node displacements according to claim 1, characterized in that: In S3, the formula for converting body load, surface load and line load into node equivalent load using the principle of virtual work is: In the formula, R i is the equivalent nodal load, N i is the unit shape function, F v is the body load distribution, F s is the surface load distribution, F l is the linear load distribution, and Ω is the corresponding distributed load integration area.

3. The method for enhancing the convergence of structural nonlinear solutions by adjusting node displacements according to claim 2, characterized in that: In S4, the constructed constraint equation is expressed as: In the formula, F i is the node displacement coefficient at the node where the load should be applied, that is, the load to be applied for the i-th degree of freedom, including node equivalent load and point load; u i is the displacement corresponding to the ith degree of freedom; u n+1 is the displacement corresponding to the main control node; is the nodal displacement coefficient at the main control node; n is the total number of nodes to which the load should be applied.

4. The method for enhancing the convergence of structural nonlinear solutions by adjusting node displacements according to claim 3, characterized in that: In S5, the process of introducing the constraint equation into the overall stiffness matrix is: S51: Obtain the degree of freedom of the main control node corresponding to the constraint equation and the degree of freedom of each node to which the load should be applied; S52: Get the maximum value K of the elements in the global stiffness matrix MAX , and define the correction coefficient P, P = 1e5*K MAX ; S53: Modify the diagonal and off-diagonal elements of the degrees of freedom corresponding to the constraint equations in the global stiffness matrix, including: a) The diagonal element K of the degrees of freedom of each node to which the load should be applied ii Add the first correction value, the first correction value is P*F i *F i ; b) The diagonal elements K of the degrees of freedom of the main control nodes jj Add a second correction value, the second correction value is c) Add a third correction value to the off-diagonal elements formed by the row corresponding to the degree of freedom of the main control node and the column j corresponding to the degree of freedom of each node to which the load should be applied. The third correction value is d) Add the third correction value to the off-diagonal elements formed by the column corresponding to the degree of freedom of the main control node and the row j corresponding to the degree of freedom of each node to which the load should be applied, that is, 5. The method for enhancing the convergence of structural nonlinear solutions by adjusting node displacements according to claim 4, characterized in that: In S6, the Newton-Raphson method is used to solve the nonlinear equations. After the solution converges, the node displacements of the structure to be analyzed at each incremental step are obtained, and the process of obtaining the displacements of the nodes to which the load should be applied based on the node displacements of the structure to be analyzed at each incremental step is as follows: S61: using the Newton-Raphson method to solve the nonlinear equations, and after the solution converges, obtaining the node displacements of the structure to be analyzed at each incremental step; S62: calculating the constraint reaction force of the structure to be analyzed at the boundary based on the node displacement of the structure to be analyzed in each incremental step, and obtaining a constraint reaction force variation curve; S63: Compare the magnitude of each value in the constraint reaction force change curve with the sum of the original load values ​​that should be applied in turn. If the displacement value of the current incremental step in the constraint reaction force change curve is less than the sum of the original load values ​​that should be applied, and the displacement value in the next incremental step is greater than the sum of the original load values ​​that should be applied, it means that the displacement value of the structure to be analyzed under the original load that should be applied corresponds to the current incremental step. Then the current incremental step is the incremental step corresponding to the original load value that should be applied. Interpolation is performed based on the displacement between the beginning and the end of the current incremental step, that is, the displacement corresponding to the node of the original load value that should be applied is obtained.

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