CN116861604A - Efficient geometric topology optimization method based on convergence acceleration - Google Patents
Efficient geometric topology optimization method based on convergence acceleration Download PDFInfo
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Abstract
The invention discloses a high-efficiency geometric topology optimization method based on convergence acceleration, which comprises two aspects of reducing design variables and gray level inhibition, wherein the number of control points needing to update density is reduced by searching the control points needing not to update density and fixing the density; the intermediate density is advanced to 0 or 1 by improving the Optimization Criteria (OC) method. The invention obviously reduces the iteration times of topology optimization, greatly improves the calculation efficiency, simultaneously enables the design result to be more approximately 0-1 distribution, and solves the problem that the traditional isogeometric topology optimization needs to give the design result through a large number of iterations, and is often difficult to meet the requirement of rapidly designing a large-scale structure.
Description
Technical Field
The invention relates to the technical field of structural topology optimization, in particular to a high-efficiency geometric topology optimization method based on convergence acceleration.
Background
Topology optimization is a structural optimization method, which refers to finding the material distribution condition when a given objective function reaches the optimum under the given constraint condition and load condition. The isogeometric analysis isogeometric topological optimization is a topological optimization method based on isogeometric analysis, and is characterized in that integration of a Computer Aided Design (CAD) model, a computer aided analysis (CAE) model and a topological optimization model is realized on a mathematical model, and compared with the traditional finite element analysis, the calculation precision is higher.
In practical engineering problems, topology optimization has the characteristic that the final configuration can be given after a plurality of iterations, so that the topology optimization is very challenging in handling large-scale engineering problems. The calculation speed of the traditional isogeometric topological optimization is difficult to meet the requirement of rapid structural design, and the search for an acceleration strategy of the isogeometric topological optimization is the key for solving the problem.
Disclosure of Invention
In order to overcome the defects and shortcomings in the prior art, the invention provides a high-efficiency isogeometric topological optimization method based on convergence acceleration.
In order to achieve the above purpose, the present invention adopts the following technical scheme:
a high-efficiency geometric topology optimization method based on convergence acceleration comprises the following steps:
s1: setting basic parameters of geometric topological optimization;
s2: performing isogeometric topology optimization pretreatment;
s3: solving an isogeometric analysis equation to obtain an overall displacement vector;
s4: calculating the structural flexibility and solving the flexibility relative to the design variable;
s5: judging whether the iteration number i is larger than the set iteration number N, if so, performing the step S6, and if not, performing the step S11;
s6: calculating the variation c of the structural flexibility obj ；
S7: judging the change c of the structural flexibility obj If the change threshold delta is smaller than the set change threshold delta, the step S8 is performed, and if the change threshold delta is larger than the set change threshold delta, the step S11 is performed;
s8: searching a control point of which the density does not need to be updated, and fixing the density of the control point;
s9: updating the design variables using an improved OC process;
s10: judging whether the convergence condition of the topology optimization is reached, if so, performing step S12, and if not, returning to step S3;
s11: updating all design variables by using an optimization criterion method, and then proceeding to step S10;
s12: and outputting the geometric topological optimization results.
As a preferable technical scheme, the basic parameters of the geometric topological optimization such as the setting include design fields, constraint conditions and load conditions of a defined structure, a defined volume fraction and a subdivided grid number.
As a preferable technical scheme, the method for performing the isogeometric topological optimization pretreatment comprises the steps of constructing a grid and calculating a unit stiffness matrix.
As a preferred embodiment, the variation c of the structural flexibility is calculated obj The specific calculation formula is expressed as:
wherein c obj Representing the variation of the structural flexibility, c represents the iterative structural flexibility, k represents the current iteration number, and N is an integer.
As a preferable technical solution, the searching for a control point that does not need to update the density refers to a control point whose density satisfies the following relationship:
max(x (k) ，x (k-1) ，...，x (k-Q+1) )＜0.01；
or min (x) (k) ，x (k-1) ，...，x (k-Q+1) )＞0.99；
Wherein x is (k) Control point density value, x, for the kth iteration (k-1) Control point density value, x, for the k-1 th iteration (k -Q+1) The control point density value of the kth Q+1 iterations, Q is an integer.
As a preferred technical solution, the fixing the density of the control point specifically includes:
will satisfy max (x (k) ，x (k-1) ，...，x (k-Q+1) ) The density of control points of < 0.01 is set to 0, while min (x (k) ，x (k-1) ，...，x (k-Q+1) ) The density of control points > 0.99 is set to 1.
As a preferable technical scheme, the improved OC method is based on a standard OC method, and the following formula is added:
wherein x' new As a final design variable, x new Is a design variable updated by a standard OC method, and a is a steepness parameter.
As a preferable technical scheme, the steepness parameter is specifically set as follows:
wherein t is a constant, c obj Indicating the amount of change in the compliance of the structure.
As an optimal technical scheme, the isogeometric topological optimization is carried out by adopting a SIMP method, the density of control points is used as a design variable, the density of units is set as the density of the central parameter coordinate points of the units, the maximum rigidity of the structure is taken as a target, the volume ratio is taken as a constraint, and an isogeometric topological optimization mathematical model is established.
As a preferred technical solution, the isogeometric topological optimization mathematical model is expressed as:
Find x＝(x 1 ，x 2 ，…x n ) T
wherein C (x) is an objective function, x e Is the unit density, x is the design variable, n is the number of design variables in the design domain, U is the overall displacement vector, F is the overall load vector, K is the overall stiffness matrix, E e The modulus of elasticity of the unit e is the following in the variable density method: e (E) e (x)＝E min +x p (E 0 -E min ) Wherein E is min Modulus of elasticity of empty units, E 0 The elastic modulus of the entity unit is obtained, and p is a punishment factor; u (u) e Is the displacement vector, k, of element e e Is the cell stiffness matrix of cell e, V (x) is the total volume of structural solid material, V 0 Is the total volume of the design domain, and VF is the volumetric ratio constraint.
Compared with the prior art, the invention has the following advantages and beneficial effects:
(1) The invention reduces the number of the calculated variables to be updated, reduces the calculated amount of the updated variable part and improves the calculation efficiency.
(2) The invention obviously reduces the iteration times of the topology optimization, thereby reducing the convergence time of the topology optimization.
(3) The optimization result of the invention is more 0-1 distribution, a large number of intermediate density units are not present, and the optimization result is more accurate.
Drawings
FIG. 1 is a flow diagram of a method for optimizing an efficient isogeometric topology based on convergence acceleration;
FIG. 2 is a final design variable x 'in the improved optimization criteria method of the present invention' new With respect to design variable x updated by standard OC method new Is a variation of the schematic diagram;
FIG. 3 is a graph showing how many geometrical topological optimization iterations and compliance before and after convergence acceleration are compared with each other at different grid scales;
FIG. 4 (a) is a schematic diagram of the optimization results of the efficient isogeometric topology optimization method of the present invention using convergence acceleration at a 30X 10X 4 grid scale;
FIG. 4 (b) is a schematic diagram of the optimization results of the efficient isogeometric topology optimization method of the present invention using convergence acceleration at a 60X 20X 6 grid scale;
fig. 4 (c) is a schematic diagram of the optimization results of the efficient isogeometric topology optimization method using convergence acceleration at a 90×30×8 grid scale according to the present invention.
Detailed Description
The present invention will be described in further detail with reference to the drawings and examples, in order to make the objects, technical solutions and advantages of the present invention more apparent. It should be understood that the specific embodiments described herein are for purposes of illustration only and are not intended to limit the scope of the invention.
Examples
As shown in fig. 1, this embodiment provides a high-efficiency geometric topology optimization method based on convergence acceleration, and is described by taking a classical Solid Isotropic Material Penalty (SIMP) method as an example, where the example uses the control point density as a design variable, and the unit density is set as the density of the unit center parameter coordinate point. Taking the maximum rigidity (namely the minimum flexibility) of the structure as a target and the volume ratio as a constraint, and establishing an isogeometric topological optimization mathematical model:
Find x＝(x 1 ，x 2 ，…x n ) T
where C (x) is an objective function, here representing the structural compliance; x is x e Is the unit density with the value range of [0,1 ]]The method comprises the steps of carrying out a first treatment on the surface of the x is a design variable, here the control point density, with a value range of [0,1]The method comprises the steps of carrying out a first treatment on the surface of the n is the number of design variables in the design domain; u is the overall displacement vector; f is the overall load vector; k is the overall stiffness matrix; e (E) e The modulus of elasticity of the unit e is the following in the variable density method: e (E) e (x)＝E min +x p (E 0 -E min ) Wherein E is min Modulus of elasticity of empty units, E 0 For the modulus of elasticity of the entity unit, p is a punishment factor, and is usually 3; u (u) e Is the displacement vector of element e; k (k) e Is the cell stiffness matrix of cell e; v (x) is the total volume of the structural solid material; v (V) 0 Is the total volume of the design domain; VF is a volume ratio constraint (VF ε [0,1 ]])；
Based on the SIMP method, the method comprises two aspects of design variable reduction and gray level suppression, the solving efficiency is improved by reducing the number of control points needing to update the density, and the intermediate density is advanced to 0 or 1 by improving an Optimization Criterion (OC) method, so that the topological optimization iteration times are reduced, and the method specifically comprises the following steps:
s1: setting basic parameters of geometric topological optimization, defining design domain, constraint condition, load condition and the like of a structure, defining volume fraction, subdivision grid number and the like, wherein the embodiment takes a traditional cantilever beam as an illustration;
s2: performing geometric topological optimization pretreatment, including grid construction, unit stiffness matrix calculation and the like;
s3: solving an isogeometric analysis equation to obtain an overall displacement vector U;
s4: calculating the structural flexibility and solving the flexibility relative to the design variable;
s5: judging whether the iteration number i is larger than the set iteration number N, if so, performing the step S6, and if not, performing the step S11;
the purpose of setting the iteration times N in the step S5 is that the step S6 needs to synthesize iteration information for multiple times to judge;
s6: calculating the variation c of the structural flexibility obj ；
In this embodiment, the amount of change in the structural flexibility is calculated by the following formula:
wherein c (i) Represents the structural flexibility of the ith iteration, k represents the current iteration number, N is an integer, and can be generallyTaking the weight of the mixture as 5-10.
S7: judging the change c of the structural flexibility obj If the change threshold delta is smaller than the set change threshold delta, the step S8 is performed, and if the change threshold delta is larger than the set change threshold delta, the step S11 is performed;
in the present embodiment, the reason why step S7 determines that the threshold δ is smaller is to reduce the influence of the reduced design variable on the result while reducing the possibility of unstable values due to gray scale suppression.
S8: searching a control point of which the density does not need to be updated, and fixing the density;
in this embodiment, step S8 does not need to update the control points of the density, which means the control points of which the density satisfies the following relationship:
max(x (k) ，x (k-1) ，...，x (k-Q+1) ) < 0.01 or min (x (k) ，x (k-1) ，...，x (k-Q+1) ) > 0.99, where x (k) For the control point density value of the kth iteration, Q is an integer, typically 5-10.
In this embodiment, step S8 fixes the density thereof, specifically:
will satisfy max (x (k) ，x (k-1) ，...，x (k-Q+1) ) The density of control points of < 0.01 is set to 0, while min (x (k) ，x (k-1) ，...，x (k-Q+1) ) The density of control points > 0.99 is set to 1.
S9: updating design variables to be updated using an improved OC method;
in this embodiment, the step S9 uses the modified OC method, which is based on the standard OC method, and the following formula is added:
wherein x' new As a final design variable, x new Is a design variable updated by a standard OC method, a is a steepness parameter, the steepness parameter a gradually becomes larger in the iterative process to improve the convergence rate, and the value is set as +.>Where t is a constant, the larger t is, the faster the convergence speed is, and the accuracy of the solution flexibility may be reduced, and the constant t in this embodiment preferably takes a value of 6;
in the present embodiment, all the design variables except for the density that has been set in step S8 are updated, and the design variable obtained by updating by the standard OC method is x new As shown in FIG. 2, x 'is obtained when different values are taken' new Concerning x new Is a variation of (2);
in the present embodiment, the design variables that need to be updated in step S9 refer to all the design variables except the design variables that do not need to be updated, which are found by step S8.
S10: judging whether the convergence condition of the topology optimization is reached, if so, performing step S12, and if not, returning to step S3;
in the present embodiment, the convergence condition is set such that the maximum variation of the design variable is less than 1% or the number of iterations is not less than 400.
S11: updating all design variables by using an optimization criterion method, and then proceeding to step S10;
s12: outputting data;
as shown in fig. 3, the comparison of the number of isogeometric topology optimization iterations before and after convergence acceleration at 30×10×4, 60×20×6, and 90×30×8 mesh scales and the flexibility are shown, and as shown in fig. 4 (a) -4 (c), the optimization results of the isogeometric topology optimization algorithm using convergence acceleration at 30×10×4, 60×20×6, and 90×30×8 mesh scales are shown, respectively. It can be seen that after convergence acceleration is used, the convergence frequency is obviously reduced, so that the solving speed is improved, and the topological configuration which is more distributed towards 0-1 is obtained through gray level suppression.
The above examples are preferred embodiments of the present invention, but the embodiments of the present invention are not limited to the above examples, and any other changes, modifications, substitutions, combinations, and simplifications that do not depart from the spirit and principle of the present invention should be made in the equivalent manner, and the embodiments are included in the protection scope of the present invention.
Claims (10)
1. The efficient isogeometric topological optimization method based on convergence acceleration is characterized by comprising the following steps of:
s1: setting basic parameters of geometric topological optimization;
s2: performing isogeometric topology optimization pretreatment;
s3: solving an isogeometric analysis equation to obtain an overall displacement vector;
s4: calculating the structural flexibility and solving the flexibility relative to the design variable;
s5: judging whether the iteration number i is larger than the set iteration number N, if so, performing the step S6, and if not, performing the step S11;
s6: calculating the variation c of the structural flexibility obj ；
S7: judging the change c of the structural flexibility obj If the change threshold delta is smaller than the set change threshold delta, the step S8 is performed, and if the change threshold delta is larger than the set change threshold delta, the step S11 is performed;
s8: searching a control point of which the density does not need to be updated, and fixing the density of the control point;
s9: updating the design variables using an improved OC process;
s10: judging whether the convergence condition of the topology optimization is reached, if so, performing step S12, and if not, returning to step S3;
s11: updating all design variables by using an optimization criterion method, and then proceeding to step S10;
s12: and outputting the geometric topological optimization results.
2. The method for optimizing the high-efficiency isogeometric topology based on convergence acceleration according to claim 1, wherein the setting of basic parameters of isogeometric topological optimization comprises defining design fields, constraint conditions and load conditions of a structure, and defining volume fraction and subdivision grid number.
3. The efficient isogeometric topological optimization method based on convergence acceleration according to claim 1, wherein the isogeometric topological optimization preprocessing comprises the steps of constructing a grid and calculating a unit stiffness matrix.
4. The method for optimizing a high-efficiency isogeometric topology based on convergence acceleration as set forth in claim 1, wherein the variation c of the structural flexibility is calculated obj The specific calculation formula is expressed as:
wherein c obj Representing the variation of the structural flexibility, c represents the iterative structural flexibility, k represents the current iteration number, and N is an integer.
5. The convergence acceleration-based efficient isogeometric topological optimization method as set forth in claim 1, wherein the searching for a control point that does not require updating the density is a control point whose density satisfies the following relationship:
max(x (k) ，x (k-1) ，...，x (k-Q+1) )＜0.01；
or min (x) (k) ，x (k-1) ，...，x (k-Q+1) )＞0.99；
Wherein x is (k) Control point density value, x, for the kth iteration (k-1) Control point density value, x, for the k-1 th iteration (k-Q+1 ) The control point density value of the k-Q+1st iteration, Q is an integer.
6. The convergence acceleration-based efficient isogeometric topological optimization method as set forth in claim 5, wherein the fixing the density of the control points specifically comprises:
will satisfy max (x (k) ，x (k-1 )，...，x (k-Q+1) ) The density of control points of < 0.01 is set to 0, while min (x (k) ，x (k-1) ，...，x (k-Q+1 ) A) the density of control points > 0.99 is set to 1.
7. The efficient isogeometric topological optimization method based on convergence acceleration as set forth in claim 1, wherein the improved OC method is based on a standard OC method, and the following formula is added:
wherein x' new As a final design variable, x new Is a design variable updated by a standard OC method, and a is a steepness parameter.
8. The efficient isogeometric topological optimization method based on convergence acceleration as set forth in claim 7, wherein the steepness parameter is specifically set as follows:
wherein t is a constant, c obj Indicating the amount of change in the compliance of the structure.
9. The efficient isogeometric topological optimization method based on convergence acceleration according to claim 1, wherein isogeometric topological optimization is performed by adopting a SIMP method, the density of control points is used as a design variable, the density of units is set as the density of the central parameter coordinate points of the units, the maximum rigidity of the structure is used as a target, the volume ratio is used as a constraint, and an isogeometric topological optimization mathematical model is established.
10. The efficient isogeometric topological optimization method based on convergence acceleration as set forth in claim 9, wherein the isogeometric topological optimization mathematical model is expressed as:
Find x＝(x 1 ，x 2 ，…x n ) T
wherein C (x) is an objective function, x e Is the unit density, x is the design variable, n is the number of design variables in the design domain, U is the overall displacement vector, F is the overall load vector, K is the overall stiffness matrix, E e The modulus of elasticity of the unit e is the following in the variable density method: e (E) e (x)＝E min +x p (E 0 -E min ) Wherein E is min Modulus of elasticity of empty units, E 0 The elastic modulus of the entity unit is obtained, and p is a punishment factor; u (u) e Is the displacement vector, k, of element e e Is the cell stiffness matrix of cell e, V (x) is the total volume of structural solid material, V 0 Is the total volume of the design domain, and VF is the volumetric ratio constraint.
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