CN108875125B - Topological optimization method for continuum dual-material structure under mixed constraint of displacement and global stress - Google Patents

Topological optimization method for continuum dual-material structure under mixed constraint of displacement and global stress Download PDF

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CN108875125B
CN108875125B CN201810391007.9A CN201810391007A CN108875125B CN 108875125 B CN108875125 B CN 108875125B CN 201810391007 A CN201810391007 A CN 201810391007A CN 108875125 B CN108875125 B CN 108875125B
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邱志平
王磊
夏海军
刘东亮
梁金雄
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Abstract

The invention discloses a topological optimization method of a continuum bi-material structure under mixed constraint of displacement and global stress, which comprises the steps of firstly describing design variables of a bi-material topological optimization problem according to a variable density method, setting two design variables for each unit, then respectively carrying out density filtering on the two design variables by using a density filtering method to obtain two pseudo densities corresponding to the units, and on the basis of a stress relaxation method, providing a bi-material stress relaxation rule to construct a stress constraint rule of the units; synthesizing the stress by utilizing a stress synthesis function to obtain a global stress constraint function, and solving the sensitivity of the global stress constraint function by utilizing an adjoint vector method and a composite function derivation method; and finally, solving the optimization problem by using a mobile progressive method, and performing iterative computation until corresponding convergence conditions are met to obtain an optimal design scheme meeting the constraint. The invention can ensure that the bi-material topological structure meeting the mixed constraint conditions of displacement and global stress is obtained, and can realize effective weight reduction.

Description

Topological optimization method for continuum dual-material structure under mixed constraint of displacement and global stress
Technical Field
The invention relates to the field of topological optimization design of a structure containing a continuum, in particular to construction of a bi-material stress constraint function and formulation of a bi-material topological optimization scheme of the continuum structure under mixed constraint of displacement and global stress.
Background
With the development of science and technology and society, people have higher and higher requirements on the performance of structures, so that the structure optimization research becomes more important. The structure optimization design can be divided into three levels: size optimization, shape optimization and topology optimization. Among them, topological optimization, which is a conceptual stage of structural design, has a decisive influence on the final configuration, and thus has the most significant influence on the structural performance. Therefore, the research and study of topological optimization design is of great significance.
The traditional topological optimization research mainly focuses on the problem that structural flexibility is taken as a target, structural volume is taken as a constraint or structural volume is taken as a target, and structural flexibility is taken as a constraint, and the research on the topological optimization problem of stress constraint is less. Stress constraints have three properties: singularities, locality, and strong non-linearities. In studying the topology optimization problem under stress constraints, gunn et al discovered the singular nature of stress and proposed the stress relaxation law. Some researchers have proposed q-p methods to solve the problem of stress singularity. Aiming at the local problem of stress constraint, some scholars propose a P-form stress comprehensive function method, and the method has a good approximate effect on the global maximum stress. Still some scholars propose a K-S stress comprehensive function method, and the sensitivity of the K-S stress comprehensive function to the design variable is solved more simply. However, the stress synthesis function method strengthens the nonlinearity of the stress constraint, so that the convergence of the solution of the optimization model is poor, and some scholars propose a method for synthesizing the stress in a sub-region aiming at the problem, so that the nonlinearity of the stress synthesis function is weakened to a certain extent.
However, it is worth noting that the current research on the topology optimization problem under stress constraint mainly aims at the single-material structure, and there is a fresh research on the dual-material or even multi-material structure. Some researchers have studied the topological optimization problem of bi-material micro-motion mechanisms under stress constraints. In the existing literature, the topological optimization problem of the continuum bi-material structure under the mixed constraint of displacement and global stress is not seen yet, however, in practical engineering, the designability of the structure is stronger due to the use of bi-material, so that the performance of the structure is further improved. Therefore, the method has important significance in researching the dual-material topology optimization problem under the mixed constraint of displacement and global stress.
Disclosure of Invention
The technical problem to be solved by the invention is as follows: the method overcomes the defects of the prior art and provides a topological optimization method of a continuum bi-material structure under the mixed constraint of displacement and global stress. The invention considers the topological optimization problem of the continuum bi-material structure under the constraints of displacement and global stress, breaks through the limitation of the traditional stress constraint topological optimization single material, and further improves the designability of the engineering structure.
The technical scheme adopted by the invention is as follows: a topological optimization method for a continuum bi-material structure under mixed constraint of displacement and global stress comprises the following implementation steps:
the method comprises the following steps: a variable density method is adopted to describe design variables, the volume of a structure is taken as an optimization target, structural displacement and global stress are taken as constraints, and a topological optimization model is established as follows:
Figure BDA0001643437480000021
where V is the volume of the optimization region, ρi(x, y) and ViRelative density and volume, respectively, of the ith cell, and piIs a function of design variables x and y, N is the total number of units divided by the optimization area, ujIs the actual displacement value, u, of the jth displacement constraint pointj,maxIs the allowable displacement value of the jth displacement constraint, m is the number of displacement constraints, σiIs the actual stress value of the ith cell,
Figure BDA0001643437480000022
to obtain the allowable stress of the material 1, i ∈ solid (1) indicates that the unit i is the material 1,
Figure BDA0001643437480000023
for the allowable stress of material 2, i ∈ solid (2) indicates that the unit i is material 2. r is a lower limit of a design variable, is a small quantity set for avoiding singularity of a stiffness matrix, and is generally 0.01;
step two: respectively filtering the design variables x and y by adopting a density filtering method to obtain two pseudo density values corresponding to the filtered unit i, and then calculating the density rho of the unit i according to the pseudo density valuesi(x, y), thereby calculating the total volume of the structure, and the sensitivity of the total volume of the structure with respect to the cell pseudo-density;
step three: calculating the elastic modulus of the unit by adopting a solid anisotropic microstructure/material model (SIMP model) double-material expansion form with penalty factors to obtain a rigidity matrix of the unit and further obtain a total rigidity matrix of the structure, then calculating the displacement of the structure according to a static equilibrium equation, establishing a double-material stress relaxation rule on the basis of a single-material stress relaxation method, constructing a stress constraint function of the unit, and adopting the Von Sauss stress at the central point of the unit as the representation of the unit stress in the stress calculation process;
step four: synthesizing all the unit stress constraint functions by adopting a P-form stress comprehensive function to obtain a stress comprehensive function, and correcting the stress comprehensive function according to the deviation of the stress comprehensive function value and the maximum value of the unit stress constraint function to obtain a global stress constraint function;
step five: solving the sensitivity of the structural displacement and the global stress constraint function to the unit pseudo-density by using an adjoint vector method, obtaining the sensitivity of the structural displacement and the global stress constraint function design variable by using a composite function derivation method, and taking a correction coefficient as a constant in the process of solving the sensitivity;
step six: taking the obtained displacement and global stress constraint function values and sensitivity information of the displacement and global stress constraint function values to design variables as input conditions of a mobile evolutionary algorithm (MMA), solving an optimization problem, and updating the design variables;
step seven: and repeating the second step to the sixth step, and updating the design variables for multiple times until the current design meets the constraint and the relative change percentage of the objective function is smaller than the preset value xi, and stopping the optimization process.
Compared with the prior art, the invention has the advantages that:
(1) the invention considers the topological optimization problem of the dual-material structure under the constraint of displacement and global stress, so that the design domain of the structure is larger, and compared with the traditional topological optimization of the single-material structure under the constraint of displacement and global stress, the invention can obtain a lighter structure;
(2) the invention provides a bi-material stress relaxation method, constructs a unit stress constraint function, and can provide a multi-material stress relaxation method on the basis, so that the content application range of the invention is wider.
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FIG. 1 is a flow chart of the topological optimization of a continuum bi-material structure under the mixed constraint of displacement and global stress according to the invention.
Detailed Description
The invention is further described with reference to the following figures and detailed description.
As shown in fig. 1, the invention provides a topological optimization method of a continuum bi-material structure under mixed constraints of displacement and global stress, which comprises the following steps:
the method comprises the following steps: a variable density method is adopted to describe design variables, the volume of a structure is taken as an optimization target, structural displacement and global stress are taken as constraints, and a topological optimization model is established as follows:
Figure BDA0001643437480000041
where V is the volume of the optimization region, ρi(x, y) and ViRelative density and volume, respectively, of the ith cell, and piIs a function of design variables x and y, N is the total number of units divided by the optimization area, ujIs the actual displacement value, u, of the jth displacement constraint pointj,maxIs the allowable displacement value of the jth displacement constraint, m is the number of displacement constraints, σiIs the actual stress value of the ith cell,
Figure BDA0001643437480000042
to obtain the allowable stress of the material 1, i ∈ solid (1) indicates that the unit i is the material 1,
Figure BDA0001643437480000043
for the allowable stress of material 2, i ∈ solid (2) indicates that the unit i is material 2. r is lower than design variableThe limit is a small amount set to avoid singularity of the stiffness matrix, and is generally 0.01;
step two: respectively filtering the design variables x and y by adopting a density filtering method to obtain two pseudo density values corresponding to the filtered unit i, and then calculating the density rho of the unit i according to the pseudo density valuesi(x, y), and then calculating the total volume of the structure;
the design variables x and y are density filtered separately in the following form:
Figure BDA0001643437480000044
wherein x'i,y′iAre two pseudo-density values, x, corresponding to the ith cellj,yjTwo design variables for the jth cell. OmegaiAll distances from the unit i are less than or equal to r0(filter radius) set of units, rjIs the distance of cell j from the center point of cell i.
The density ρ of the cell ii(x, y) can be solved using the following equation:
ρi(x,y)＝x′i[y′iρ1+(1-y′i)ρ2]
where ρ is1Is the density of the material 1, p2Is the density of material 2.
The total volume of the structure can be calculated as:
Figure BDA0001643437480000045
wherein, ViIs the unit volume of the ith unit.
In particular, the total volume of the structure versus the pseudo density x 'of the ith cell'i,y′iThe derivative of (d) can be calculated as:
Figure BDA0001643437480000051
Figure BDA0001643437480000052
step three: calculating the elastic modulus of the unit by adopting a solid anisotropic microstructure/material model (SIMP model) double-material expansion form with penalty factors to obtain a rigidity matrix of the unit and further obtain a total rigidity matrix of the structure, then calculating the displacement of the structure according to a static equilibrium equation, establishing a double-material stress relaxation rule on the basis of a single-material stress relaxation method, constructing a stress constraint function of the unit, and adopting the Von Sauss stress at the central point of the unit as the representation of the unit stress in the stress calculation process;
the elastic modulus of unit i is calculated using a two-material extension version of the SIMP model:
Figure BDA0001643437480000053
wherein E is1Is the modulus of elasticity, E, of the material 12Is the modulus of elasticity of material 2.
After the unit elastic modulus is obtained, a unit rigidity matrix can be constructed, so that an overall rigidity matrix of the structure is obtained, and static calculation is carried out to obtain the displacement of the unit.
To better characterize the stress level of the structure, von mises stress was used to characterize the cell stress. The mathematical expression is as follows:
Figure BDA0001643437480000054
wherein σ1,σ2,σ3Respectively, the first, second and third principal stresses.
According to the obtained displacement of the unit node and the corresponding displacement shape function and the strain matrix, the normal stress and the shear stress of each direction of the unit can be obtained as follows:
σ＝D＝DBue＝Sue
in the formula, the elastic modulus of the cell strain matrix is the unit elastic modulus after penalty.
In order to simplify the calculation and to take into account that the stress variation within the cell is small, the stress at the center point of the cell is taken as an indication of the cell stress.
The Von Mileiser stress at the central point can be obtained according to the normal stress and the shear stress of the central point of the unit in all directions as follows:
Figure BDA0001643437480000055
wherein σc＝[σcx,σcy,σcz,τaxy,τayz,τazx]TThe load column vector at the center point of the cell.
For convenience of description, note:
σcr＝h(σc)
after the unit stress is obtained, a two-material stress relaxation law is constructed on the basis of the single-material stress relaxation law.
For a single material stress constraint topology optimization problem, for unit i, the original constraint is:
Figure BDA0001643437480000061
it is rewritten as:
Figure BDA0001643437480000062
this equation is equivalent to:
Figure BDA0001643437480000063
constraint relaxation is performed as follows:
Figure BDA0001643437480000064
rewritten again as:
Figure BDA0001643437480000065
on the basis of a single-material stress relaxation method, the two-material stress constraint function of the structural unit i is as follows:
Figure BDA0001643437480000066
wherein (sigma)i)crVon mises stress at the ith cell center point,
Figure BDA0001643437480000067
for the allowable stress of the material 1,
Figure BDA0001643437480000068
for the allowable stress of the material 2, for the small amount of stress relaxation, there are2＝r。
Step four: synthesizing all the unit stress constraint functions by adopting a P-form stress comprehensive function to obtain a stress comprehensive function, and correcting the stress comprehensive function according to the deviation of the stress comprehensive function value and the maximum value of the unit stress constraint function to obtain a global stress constraint function;
and integrating all unit stress constraint functions by adopting a stress integration function in a P form. Considering that the unit stress constraint function may be less than zero, the global stress constraint function is constructed as follows:
Figure BDA0001643437480000071
wherein, P is a stress synthesis index, and the higher P represents the better effect of the stress synthesis function on the approximation of the global maximum stress, but the increase of P also increases the nonlinearity of the stress synthesis function, so that the convergence of the problem is poor. P can be 6-12, and is 8 or 10.
c is a stress correction coefficient, and the global stress constraint function is corrected according to the unit stress constraint function of the maximum stress unit. The unit stress constraint function of the maximum stress unit is as follows:
Figure BDA0001643437480000072
therefore, the correction coefficient c is expressed as follows:
Figure BDA0001643437480000073
when the sensitivity solution of the global stress constraint function is performed below, c is treated as a constant.
The global stress constraint is:
Φ-1≤0
step five: solving the sensitivity of the structural displacement and the global stress constraint function to the unit pseudo-density by using an adjoint vector method, obtaining the sensitivity of the structural displacement and the global stress constraint function design variable by using a composite function derivation method, and taking a correction coefficient as a constant in the process of solving the sensitivity;
since the sensitivity of structural displacements to design variables has been derived in some literature, only the global stress function versus the cell pseudo density x 'is derived below'j,y′jThe sensitivity of (2).
Global stress function pair x'jThe sensitivity of (d) is solved as:
Figure BDA0001643437480000081
wherein
Figure BDA0001643437480000082
The solution is as follows:
Figure BDA0001643437480000083
due to (sigma)cr)iIs about x'i,y′iAnd u, for convenience of explanation, are
Figure BDA0001643437480000084
Then
Figure BDA0001643437480000085
The following can be solved:
Figure BDA0001643437480000086
order to
Figure BDA0001643437480000087
Figure BDA0001643437480000088
Figure BDA0001643437480000091
Then there are:
Figure BDA0001643437480000092
finishing is carried out, and the method comprises the following steps:
Figure BDA0001643437480000093
the first item of
Figure BDA0001643437480000094
Can be directly calculated from the previous result, the second term
Figure BDA0001643437480000095
It is necessary to obtain the vector by the adjoint vector method. Setting:
Figure BDA0001643437480000096
wherein the adjoint vector
Figure BDA0001643437480000097
Can be obtained by a finite element calculation. Then
Figure BDA0001643437480000101
Can be solved as:
Figure BDA0001643437480000102
in the formula, KjIs a unit stiffness matrix of the unit j, and u is the structural displacement under the actual working condition. To this end, a global stress function is obtained with respect to a unit design variable x'jThe sensitivity of (2).
Similarly, global stress function pair y'jThe sensitivity of (d) is solved as:
Figure BDA0001643437480000103
wherein
Figure BDA0001643437480000104
The solution is as follows:
Figure BDA0001643437480000105
in the formula (I), the compound is shown in the specification,
Figure BDA0001643437480000106
the solution is as follows:
Figure BDA0001643437480000111
order to
Figure BDA0001643437480000112
Then there are:
Figure BDA0001643437480000113
finishing is carried out, and the method comprises the following steps:
Figure BDA0001643437480000114
the first item of
Figure BDA0001643437480000115
Can be directly calculated from the previous result, the second term
Figure BDA0001643437480000116
It is necessary to obtain the vector by the adjoint vector method. Setting:
Figure BDA0001643437480000121
wherein the adjoint vector
Figure BDA0001643437480000122
Can be obtained by a finite element calculation. Then
Figure BDA0001643437480000123
Can be solved as:
Figure BDA0001643437480000124
by this, the sensitivity solution of the global stress function with respect to the cell pseudo-density is completed.
The sensitivity of the cell pseudo-density with respect to design variables is solved below.
Density x 'of unit i'iDesign variable x for cell jjThe sensitivity of (d) is given by:
Figure BDA0001643437480000125
wherein omegaiAll distances from the unit i are less than or equal to r0(filter radius) set of units, rjIs the distance of cell j from the center point of cell i.
Similarly, density y 'of unit i'iDesign variable y for cell jjThe sensitivity of (d) is given by:
Figure BDA0001643437480000126
wherein omegaiAll distances from the unit i are less than or equal to r0(filter radius) set of units, rjIs the distance of cell j from the center point of cell i.
Thus, sensitivity of cell pseudo-density to cell design variations is obtained.
Applying the complex function derivation rule, there are:
Figure BDA0001643437480000127
in a similar way, the method comprises the following steps:
Figure BDA0001643437480000131
thus, the sensitivity of the global stress function with respect to design variables is obtained.
Step six: taking the obtained displacement and global stress constraint function values and sensitivity information of the displacement and global stress constraint function values to design variables as input conditions of a mobile evolutionary algorithm (MMA), solving an optimization problem, and updating the design variables;
step seven: and repeating the second step to the sixth step, and updating the design variables for multiple times until the current design meets the constraint and the relative change percentage of the objective function is smaller than the preset value xi, and stopping the optimization process.
The invention provides a topological optimization method of a continuum bi-material structure under mixed constraint of displacement and global stress. Firstly, establishing a continuum bi-material structure topological optimization model which takes the minimum structure weight as an optimization target and takes structure displacement and global stress as constraints; then, a density filtering method is used for filtering unit design variables respectively to obtain unit pseudo density, the elastic modulus of the unit is calculated in a double-material expansion form of an SIMP model, the displacement of the structure is calculated by using a finite element, a unit stress constraint function is constructed by using a double-material stress relaxation method, a global stress constraint function is constructed by using a stress comprehensive function method, and the sensitivity of the global stress constraint function is solved by using an adjoint vector method and a composite function derivation method; and finally, carrying out optimization solution by using a mobile progressive method until corresponding convergence conditions are met, and obtaining a topological optimization design scheme meeting displacement and global stress constraints.
The above are only the specific steps of the present invention, and the protection scope of the present invention is not limited in any way; the method can be expanded and applied to the field of topological optimization design of a continuous body bi-material structure under the mixed constraint of displacement and global stress, and all technical schemes formed by adopting equivalent transformation or equivalent replacement fall within the protection scope of the invention.
The invention has not been described in detail and is part of the common general knowledge of a person skilled in the art.

Claims (1)

1. A topological optimization method for a continuum bi-material structure under mixed constraint of displacement and global stress is characterized by comprising the following steps: firstly, establishing a continuum bi-material structure topology optimization model which takes the minimum structure weight as an optimization target and takes the structure displacement and the global stress as constraints; then, a density filtering method is used for filtering unit design variables respectively to obtain unit pseudo density, the elastic modulus of the unit is calculated in a double-material expansion form of an SIMP model, the displacement of the structure is calculated by using a finite element, a unit stress constraint function is constructed by using a double-material stress relaxation method, a global stress constraint function is constructed by using a stress comprehensive function method, and the sensitivity of the global stress constraint function is solved by using an adjoint vector method and a composite function derivation method; and finally, carrying out optimization solution by using a mobile progressive method until corresponding convergence conditions are met, and obtaining a topology optimization design scheme meeting displacement and global stress constraints, wherein the implementation steps are as follows:
the method comprises the following steps: a variable density method is adopted to describe design variables, the volume of a structure is taken as an optimization target, structural displacement and global stress are taken as constraints, and a topological optimization model is established as follows:
Figure FDA0002574776570000011
where V is the volume of the optimization region, ρi(x, y) and ViRelative density and volume, respectively, of the ith cell, and piIs a function of design variables x and y, N is the total number of units divided by the optimization area, ujIs the actual displacement value, u, of the jth displacement constraint pointj,maxIs the allowable displacement value of the jth displacement constraint, m is the number of displacement constraints, σiIs the actual stress value of the ith cell,
Figure FDA0002574776570000012
to obtain the allowable stress of the material 1, i ∈ solid (1) indicates that the unit i is the material 1,
Figure FDA0002574776570000013
for the allowable stress of the material 2, i belongs to solid (2) and represents that the unit i is the material 2, r is the lower limit of a design variable, is a small quantity set for avoiding singularity of a stiffness matrix, and is 0.01;
step two: respectively filtering the design variables x and y by adopting a density filtering method to obtain two pseudo density values corresponding to the filtered unit i, and then calculating the density rho of the unit i according to the pseudo density valuesi(x, y), and then calculating the total volume of the structure;
step three: calculating the elastic modulus of the unit by adopting a solid each-way microstructure/material SIMP model bi-material expansion form with penalty factors to obtain a rigidity matrix of the unit, further obtaining an overall rigidity matrix of the structure, then calculating the displacement of the structure according to a static equilibrium equation, and then constructing a stress constraint function of the unit by using a bi-material stress relaxation rule;
step four: synthesizing all the unit stress constraint functions by adopting a P-form stress comprehensive function to obtain a stress comprehensive function, and correcting the stress comprehensive function according to the deviation of the stress comprehensive function value and the maximum value of the unit stress constraint function to obtain a global stress constraint function;
step five: solving the sensitivity of the structural displacement and the global stress constraint function to the design variable by using a composite function derivative rule and an adjoint vector method, and taking a correction coefficient as a constant in the process of solving the sensitivity;
step six: taking the obtained displacement and global stress constraint function values and sensitivity information of the displacement and global stress constraint function values to design variables as input conditions of a mobile asymptote method MMA, solving an optimization problem, and updating the design variables;
step seven: repeating the second step to the sixth step, and updating the design variables for multiple times until the current design meets the constraint and the relative change percentage of the objective function is smaller than the preset value xi, and stopping the optimization process;
in the second step, a density filtering method is adopted to respectively filter the two groups of design variables to obtain two pseudo density values corresponding to each unit, and further obtain the density and the volume of the unit;
in the third step, a unit stress constraint function is constructed by applying a dual-material stress relaxation rule;
in the fourth step, a stress comprehensive function in a P form is adopted to synthesize all unit stresses to obtain a bi-material global stress constraint function;
solving the sensitivity of the displacement of the dual-material structure and the global stress constraint function to the design variable according to a composite function derivation rule and an adjoint vector method;
and in the sixth step, the MMA algorithm is used for solving the topological optimization problem of the continuum structure bi-material under the mixed constraint of displacement and global stress.
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