CN111159946B - Discontinuous problem partition solving method based on minimum potential energy principle - Google Patents
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Abstract
The invention discloses a discontinuous problem partition solving method based on a minimum potential energy principle, which comprises the following steps of: 1) determining an object, establishing a discrete model, and inputting related data; 2) calculating an external load increment; 3) assembling a rigidity matrix, a flexibility matrix and an interface control matrix according to the subareas; 4) assembling a load array; 5) solving a finite element equation according to the partition; 6) iteratively solving the contact force; 7) judging the convergence condition of the load increment step; 8) and outputting a load step result and judging the incremental iteration completion condition. The advantages are that: the invention can solve the problems of discontinuous structure in engineering, low numerical simulation precision and high convergence difficulty when a contact surface exists, and has better convergence and more efficient numerical simulation process for solving the problem of discontinuous structure.
Description
Technical Field
The invention relates to a discontinuous problem partition solving method based on a minimum potential energy principle, and belongs to the technical field of numerical calculation of engineering nonlinear contact problems.
Background
The artificial structure and the natural rock mass in the engineering have high complexity and specificity, and a large amount of linear and nonlinear analysis is needed to obtain the response of the structure under various load working conditions. When the elastoplasticity, the damage and the damage process of the whole large-scale structure are analyzed, the problem of solving a large-scale or ultra-large-scale nonlinear system is solved. The solution of the nonlinear problem needs to continuously correct the rigidity matrix of the structure and adopt smaller calculation step length, which brings difficulty to practical application. When extreme state analysis is carried out, the critical structure is damaged, so that plastic damage, damage or overall instability are caused; while the remaining part of the member is still in an elastic state. In the solving process, the integral non-linear solving is carried out due to the fact that most systems in the linear stage are included, and a large amount of computing resources are wasted.
In numerical calculations, in order to obtain accurate calculation results, it is necessary to simulate the contact characteristics of different interfaces. The Zhao Lanhao published in the geotechnical engineering journal of Zhao Lanhao "finite element hybrid method with initial gap friction contact problem", the behavior of transverse seam in dynamic analysis of arch dam is simulated, but because the rigid body displacement of the block body cannot be considered, the situation that a plurality of seam surfaces are cut in a staggered way and generate a plurality of separation bodies is difficult to simulate; li Tongchun published An article of An interactive method of interface boundary elements and partial fine elements for local connection/disconnection formation schemes in International Journal for numerical Methods in Engineering, proposes a partition finite element and An interface element solution, introduces rigid body displacement, can consider the problem of multi-body contact of elastic conditions, but because of nonlinear coupling of contact and material, the interface equation flexibility coefficient is difficult to determine, and meanwhile, the method establishes An interface iteration equation from contact surface force balance, does not consider contact surface energy conservation, and has larger difficulty in calculation convergence.
Disclosure of Invention
The invention aims to overcome the defects of the prior art and provides a discontinuous problem partition solving method based on a minimum potential energy principle.
In order to solve the technical problem, the invention provides a discontinuous problem partition solving method based on a minimum potential energy principle, which comprises the following steps of:
step 1, acquiring a structure model containing a discontinuous contact surface and related data of the structure model, and dividing the structure model into a finite element mesh model;
step 2, calculating the external load increment through the related data, and determining the step number of the load step corresponding to the external load increment at the moment;
step 3, assembling a rigidity matrix k according to the partition condition in the finite element grid model i A flexibility matrix C and an interface control matrix A;
step 4, assembling a block load array delta R according to the external load increment i ；
step 6, solving a control interface equation according to a finite element solution result to obtain a contact force, accumulating the contact force to a block load array, solving the finite element equation again according to partitions, judging whether the contact is converged, and if so, entering step 7; if not, substituting the contact force into a control interface equation, and repeating the step 6 for iteration;
step 7, carrying out convergence judgment on the load increment step to obtain a residual error, judging the load increment step to converge if the residual error is smaller than a preset value, and entering step 8; if not, adding the residual error to the load array in the step 4, and then performing the step 5;
step 8, outputting a calculation result of the load step, judging whether incremental iteration is finished or not, and if so, finishing the calculation; and if not, adding the external load increment obtained by calculation in the step 2 on the basis of the current external load, adding 1 to the number of steps of the load step, and then performing the step 4.
Further, in step 1, the structural model of the discontinuous contact surface comprises information of the block and the contact surface; the related data comprises load information, loading step length, constraint information and material information.
Further, in step 2, the external load increment is a value added each time in the external load step-by-step loading process.
Further, in step 3, the stiffness matrix k i Respectively assembling according to block partitions;
the compliance matrix C ═ k -1 Solving by adopting a unit force method, wherein the block nodes outside the contact surface do not participate in the operation, and k represents a stiffness matrix;
the interface control matrix
Where ω is a transformation matrix, and for any node i (x, y, z), the transformation matrix is:
wherein (x) g ,y g ,z g ) Is a rigid body centroid coordinate.
Further, in step 5, the finite element equation is:
∑[k i Δu i ]＝∑[ΔR i ] (2)
wherein, Δ R i For bulk loaded arrays, Δ u i The finite element model node displacement is represented.
Further, in step 6, according to the principle of minimum potential energy, the total potential energy of the whole system is as follows:
wherein b represents all blocks in the structural model, Ω represents the solution area in the structural model, and σ represents ij Stress tensor, epsilon, representing a model of a structure ij Strain tensor, Ω, representing a structural model b The internal region of the block is shown,
the contact surface of the block is shown,
show justThe resultant external force applied to the body-shaped core points,
representing elastic deformation of the block, gamma i Representing a rigid body displacement of the centroid of the block,
indicating contact force on contact surface of block, u i A displacement vector representing the model of the structure,
representing physical forces in the structural model, s represents boundaries,
representing the area of the block surface subjected to surface loading;
considering the standing value problem of functional pi, determining a contact surface function constraint condition:
wherein G is an interface function, G (u) is a relative variation of a contact gap between contact surfaces of the blocks,
expressing the characteristic flexibility coefficient of the contact surface material; f represents contact force, and a Lagrange multiplier method is utilized to introduce constraint conditions into the potential energy functional to obtain a correction functional:
after the finite element is dispersed, the functional is as follows:
the interface control equation is obtained according to the minimum potential energy principle, and after the interface control equation is dispersed by using a weighted augmented Lagrange multiplier method, the expression is as follows:
wherein, gamma is rigid displacement; i is an identity matrix; f is a single-unit-force array; lambda is the contact force to be solved; f is lambda, u obtained by iteration of the previous step e Representing the deformation displacement of the block;
the contact convergence condition judgment is controlled by a preset contact calculation residual error allowable value.
Further, in step 7, the convergence condition of the increment step is determined by a preset increment step residual tolerance value.
Further, in step 8, the iteration completion condition of the external load increment is controlled by a preset calculation step length.
Further, when the contact surface is a cross contact surface, the contact surface control equation becomes:
wherein: p is the penalty function matrix and T represents the matrix transpose.
The invention achieves the following beneficial effects:
the invention utilizes the principle of minimum potential energy to establish a discontinuous interface control equation, deduces a partitioned finite element solution considering contact nonlinearity and material nonlinearity, assembles a rigidity matrix without assembling a large rigidity matrix containing all blocks, assembles a plurality of small rigidity matrices, and only modifies the nonlinear part in the solving process; the method provided by the invention can improve the calculation precision and reduce the whole calculation amount; when a plurality of contact surfaces are crossed, the method can effectively improve the condition number of the contact surface control equation matrix and enhance the solving stability; in the stability problem solving process, when structural instability occurs and a general numerical calculation method cannot converge, the method provided by the invention is easy to converge and can provide a more reasonable safety coefficient.
Drawings
FIG. 1 is a flow chart of a method of the present invention;
FIG. 2 is a diagram of the cantilever beam elasto-plastic zoning;
FIG. 3 is a schematic illustration of a cantilever damage distribution;
FIG. 4 is a graph comparing vertical displacement calculations for a cantilever beam.
Detailed Description
In order to make the objects, features and advantages of the present invention more obvious and understandable, the technical solutions in the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings in the embodiments of the present invention. All other embodiments, which can be obtained by a person skilled in the art without making any creative effort based on the embodiments in the present invention, belong to the protection scope of the present invention.
The technical scheme of the invention is further explained by the specific implementation mode in combination with the attached drawings.
The invention provides a discontinuous problem partition solving method based on a minimum potential energy principle. The invention provides a discontinuous problem partition solving method based on a minimum potential energy principle, which is based on the minimum potential energy principle. The specific implementation flow chart is shown in the attached figure 1. Mainly comprises 8 steps: (1) determining an object, establishing a discrete model, and inputting related data; (2) calculating the increment of the external load; (3) assembling stiffness matrix k according to partitions i Assembling a flexibility matrix C and an interface control matrix A; (4) assembled load array delta R i (ii) a (5) Solving a finite element equation according to the partition; (6) solving a control interface equation to obtain a contact force f, accumulating the contact force to obtain a block load array, solving a finite element equation again according to partitions, judging whether the contact is converged, and if so, entering a step 7; if not, substituting the contact force f into the equation, and repeating the step 6Iteration is carried out; (7) judging whether the load increment step is converged, if so, entering a step 8; if not, returning to the step 4; (8) outputting a load step result, judging whether incremental iteration is finished or not, and if so, finishing the calculation; if not, returning to the step 2.
The following is described in detail according to a flow chart:
step one, determining a cantilever beam as an object, wherein an elastic-plastic partition diagram of the cantilever beam is shown in figure 2;
the cantilever beam has a size of 30m × 3m × 1m and is subjected to uniform load P equal to 10 4 N/m 2 Acting, the left end is fixed with a support, and the influence of dead weight is not counted; the material parameters are as follows: elastic modulus E is 21GPa, Poisson's ratio v is 0.2, density rho is 2.4 multiplied by 10 3 kg/m 3 ；
Secondly, calculating an external load increment, wherein the external load increment is a value increased every time in the external load step-by-step loading process, and the smaller the external load increment is, the more accurate the calculation result is, the better the convergence is and the more the increment iteration times are; in this example, the number of incremental iteration steps is defined as 10 steps, and the increment of the external load is defined as 10 3 N/m 2 ；
Thirdly, assembling a rigidity matrix k according to partitions i Assembling a flexibility matrix C and an interface control matrix A;
stiffness matrix k i Respectively assembling according to block partitions;
compliance matrix C ═ k -1 Solving by adopting a unit force method, and the block nodes outside the contact surface do not participate in the operation;
interface control matrix
Where ω is a transformation matrix, and for any node i (x, y, z), the transformation matrix is:
wherein: (x) g ,y g ,z g ) Is a rigid body centroid coordinate.
Fourth, assembling the load arrayΔR i ；
Step five, solving a finite element equation according to the subareas, wherein the finite element equation is as follows:
∑[k i Δu i ]＝∑[ΔR i ] (2)
solving a control interface equation to obtain a contact force f;
the total potential energy of the whole system is as follows:
the interface control equation is obtained according to the minimum potential energy principle, and after the dispersion is carried out by utilizing a weighted augmented Lagrange multiplier method, the expression is as follows:
wherein the content of the first and second substances,
the coefficient of the flexibility of the contact surface material characteristic is obtained;
gamma is rigid displacement; i is an identity matrix; f is a single-unit-force array;
lambda is the strip solved contact force; f is lambda obtained by the previous step of iteration.
Accumulating the contact force on the block load array, solving the finite element equation again according to the partition, judging whether the contact is converged, and if so, entering the seventh step; if not, substituting the contact force f into the equation, and repeating the sixth step for iteration;
seventhly, judging whether the load increment step is converged or not, and if yes, entering the eighth step; if not, returning to the fourth step;
step eight, outputting a load step result, judging whether incremental iteration is finished or not, and if so, finishing the calculation; if not, returning to the second step to continue the load increment loading.
A schematic diagram of the distribution of cantilever damage is shown in figure 3.
Comparing the grid computing examples with different accuracies of the cantilever beam with an integral solving mode, and comparing the partition solving time of the cantilever beam, as shown in the following table 1:
TABLE 1
The comparison of the vertical displacement calculation results of the cantilever beam is shown in fig. 4, and the solution precision can be ensured by adopting a partition solution method. Therefore, by using the partition solving method provided by the invention, the solution is carried out when the number of model units is large, and more solver time, namely corresponding computing resources, can be saved under the condition of ensuring the solving precision and convergence performance.
It will be evident to those skilled in the art that the invention is not limited to the details of the foregoing illustrative embodiments, and that the present invention may be embodied in other specific forms without departing from the spirit or essential attributes thereof. The present embodiments are therefore to be considered in all respects as illustrative and not restrictive, the scope of the invention being indicated by the appended claims rather than by the foregoing description, and all changes which come within the meaning and range of equivalency of the claims are therefore intended to be embraced therein. Any reference sign in a claim should not be construed as limiting the claim concerned.
The above-mentioned embodiments are only used for illustrating the technical solutions of the present invention, and not for limiting the same; although the present invention has been described in detail with reference to the foregoing embodiments, it will be understood by those of ordinary skill in the art that: the technical solutions described in the foregoing embodiments may still be modified, or some technical features may be equivalently replaced; and such modifications or substitutions do not depart from the spirit and scope of the corresponding technical solutions of the embodiments of the present invention.
Claims (8)
1. A discontinuous problem partition solving method based on a minimum potential energy principle is characterized by comprising the following steps:
step 1, acquiring a structure model containing a discontinuous contact surface and related data of the structure model, and dividing the structure model into a finite element mesh model; the related data comprises load information, loading step length, constraint information and material information;
step 2, calculating the external load increment through the related data, and determining the step number of the load step corresponding to the external load increment at the moment;
step 3, assembling a rigidity matrix k according to the partition condition in the finite element grid model i A flexibility matrix C and an interface control matrix A;
the stiffness matrix k i Respectively assembling according to block partitions;
k is the compliance matrix C i -1 Solving by adopting a unit force method, and the block nodes outside the contact surface do not participate in the operation;
the interface control matrix
Where ω is a transformation matrix, and for any node i (x, y, z), the transformation matrix is:
wherein (x) g ，y g ，z g ) Is a rigid body centroid coordinate;
step 4, assembling a block load array delta R according to the external load increment i ；
Step 5, according to the rigidity matrix k i Flexibility matrix C, interface control matrix A and block load array delta R i Respectively solving finite element equations according to the partition conditions in the finite element mesh model, and outputting finite element solution results;
step 6, solving a control interface equation according to a finite element solution result to obtain a contact force, accumulating the contact force to a block load array, solving the finite element equation again according to partitions, judging whether the contact is converged, and if so, entering step 7; if not, substituting the contact force into a control interface equation, and repeating the step 6 for iteration;
step 7, carrying out convergence judgment on the load increment step to obtain a residual error, judging the load increment step to be converged if the residual error is smaller than a preset value, and entering step 8; if not, adding the residual error to the load array in the step 4, and then performing the step 5;
step 8, outputting a calculation result of the load step, judging whether incremental iteration is finished or not, and if so, finishing the calculation; and if not, adding the external load increment obtained by calculation in the step 2 on the basis of the current external load, adding 1 to the number of steps of the load step, and then performing the step 4.
2. The method for solving the partition of the discontinuity problem based on the minimum potential energy principle according to claim 1, wherein in step 1, the structural model of the discontinuous contact surface comprises information of a block and the contact surface.
3. The method for solving the discontinuous problem partition based on the minimum potential energy principle according to claim 1, wherein in the step 2, the external load increment is a value which is increased every time in the external load step-by-step loading process.
4. The method for solving the partition of the discontinuity problem based on the minimum potential energy principle as claimed in claim 1, wherein in step 5, the finite element equation is:
∑[k i Δu i ]＝∑[ΔR i ] (2)
wherein, Δ R i For bulk loaded arrays, Δ u i The finite element model node displacement is represented.
5. The method for solving the discontinuous problem partition based on the minimum potential energy principle as claimed in claim 4, wherein in the step 6, according to the minimum potential energy principle, the total potential energy of the whole system is as follows:
wherein b represents all blocks in the structural model, omega represents the solution area in the structural model, and sigma represents ij Stress tensor, epsilon, representing a model of a structure ij Strain tensor, Ω, representing structural model b The internal region of the block is shown,
showing the contact surface of the block or blocks,
representing the resultant external force applied to the rigid centroid,
representing elastic deformation of the block, gamma i Representing a rigid body displacement of the centroid of the block,
indicating contact force on contact surface of block, u i A displacement vector representing the model of the structure,
representing physical forces in the structural model, s represents boundaries,
representing the area of the block surface subjected to surface loading;
considering the standing value problem of functional pi, determining a contact surface function constraint condition:
wherein G is an interface function, G (u) is a relative variation of a contact gap between contact surfaces of the blocks,
expressing the characteristic flexibility coefficient of the contact surface material; f represents contact force, and a Lagrange multiplier method is utilized to introduce constraint conditions into the potential energy functional to obtain a correction functional:
after the finite element is dispersed, the functional is as follows:
the interface control equation is obtained according to the minimum potential energy principle, and after the interface control equation is dispersed by using a weighted augmented Lagrange multiplier method, the expression is as follows:
wherein, gamma is rigid displacement; i is an identity matrix; f is a unit force array; lambda is the contact force to be solved; f is lambda, u obtained by iteration of the previous step e Representing the deformation displacement of the block;
the contact convergence condition is judged by presetting the contact calculation residual error allowable value control.
6. The method for partition solution of discontinuity problem based on minimum potential energy principle according to claim 1, wherein in step 7, the convergence condition judgment of said increment step is controlled by a preset increment step residual tolerance value.
7. The method for partition solution of a discontinuity problem according to the minimum potential energy principle of claim 1, wherein in step 8, the iteration completion of the external load increment is controlled by a preset calculation step length.
8. The method for solving the discontinuous problem partition based on the minimum potential energy principle of claim 5, wherein when the contact surface is a cross contact surface, the contact surface control equation is changed into:
wherein: p is the penalty function matrix and T represents the matrix transpose.
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| CN103838913A (en) * | 2013-12-25 | 2014-06-04 | 广西科技大学 | Finite element method for curve box girder curved bridge |
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