CN112765856B - Mixed level set method for topological optimization of functionally graded porous structure - Google Patents

Mixed level set method for topological optimization of functionally graded porous structure Download PDF

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CN112765856B
CN112765856B CN202110090442.XA CN202110090442A CN112765856B CN 112765856 B CN112765856 B CN 112765856B CN 202110090442 A CN202110090442 A CN 202110090442A CN 112765856 B CN112765856 B CN 112765856B
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付君健
舒正涛
田启华
杜义贤
周祥曼
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Abstract

The invention discloses a mixed level set method for topological optimization of a functional gradient porous structure. Defining a global weight coefficient as a design variable, taking the structural rigidity performance as an objective function and the volume fraction of the porous structure as a constraint condition, and establishing a functional gradient porous structure topological optimization mathematical model based on a mixed level set method. And carrying out iterative solution of a gradient-based optimization algorithm by deducing the sensitivity of the objective function and the constraint condition on the design variable, and obtaining the porous structure with the geometry and the function in the spatial gradient distribution when the iterative process meets the constraint condition. The method provided by the invention can be combined with a mature gradient optimization algorithm, has rigorous mathematical theory derivation, can avoid the scale separation problem of a functional gradient porous structure, and improves the continuity between unit cells.

Description

Mixed level set method for topological optimization of functionally graded porous structure
Technical Field
The invention belongs to the technical field related to digital design, and particularly relates to a mixed level set method for topological optimization of a functionally graded porous structure.
Background
The functional gradient porous structure is a porous structure with space change property, the mechanical property of the functional gradient porous structure shows gradient change along a certain direction, and the mechanical property which is not possessed by the conventional uniform structural material can be met, such as higher mechanical strength, bending resistance, bearing energy absorption characteristics and the like, so that the functional gradient porous structure is widely applied to various aspects of aerospace, bionic design and the like. The traditional functionally gradient porous structure material has a single distribution form and cannot meet the requirements of complex engineering environments. In the optimization design method for the functional gradient porous structure, part of the optimization methods control the volume fraction of the porous structure based on the stress distribution of finite element analysis, so that the performance of the functional gradient porous structure is improved, but the optimal design cannot be achieved.
In recent years, it has become a trend to optimize and design a functionally graded porous structure by using a topological optimization method, for example, the porous structure is mapped onto a unit density according to a volume fraction to generate a numerically optimal functionally graded porous structure, but the macro configuration design of such an optimization method is separated from the unit configuration design, and the performance of the porous structure cannot be fully exerted. In addition, the topological optimization method based on the numerical homogenization method is widely applied to the optimization design of the functional gradient porous structure due to the advantages of strict theoretical derivation, mathematical formulas and the like, and in practical application, the homogenization method is more suitable for the topological optimization of the small-scale, large-scale and single-type periodic porous structure, and the scale separation characteristic of the homogenization method is difficult to ensure the connectivity among the porous structures. In order to avoid the scale separation problem of the functional gradient porous structure and improve the continuity between unit cells, the invention provides a mixed level set method for topological optimization of the functional gradient porous structure.
Disclosure of Invention
Aiming at the defects or improvement requirements of the prior art, the invention provides a mixed level set method for topological optimization of a functionally graded porous structure, which is characterized in that a mixed level set function of the functionally graded porous structure is constructed by adopting an implicit level set function to carry out geometric description based on two porous structure unit cell configurations, and a porous structure with the geometric and functional gradient distribution in space is obtained by solving through a gradient optimization algorithm with the rigidity maximized as a target. The mixed level set method for topological optimization of the functional gradient porous structure can avoid the scale separation problem of the functional gradient porous structure, improve the continuity among unit cells and has rigorous mathematical theory derivation.
In order to achieve the above object, the present invention provides a mixed level set method for topological optimization of a functionally graded porous structure, comprising the steps of:
s1: based on two porous structure single cell configurations, an implicit level set function is adopted for geometric description to construct a single cell mixed level set function of a porous structure;
s2: defining a plurality of mixed level set functions in a space coordinate system, wherein each mixed level set function represents a porous structure unit cell;
s3: performing space-time decoupling on a Hamilton-Jacobian partial differential equation for controlling the evolution of the level set function, and parameterizing the mixed level set function;
s4: defining a finite element mesh in a space coordinate system, and matching an implicit geometric model with a physical model;
s5: defining a global design variable on the vertex of each porous structure unit cell, and defining a local design variable inside the unit cell;
s6: establishing a functional gradient porous structure topological optimization mathematical model based on a mixed level set method by taking the structural rigidity performance as an objective function and the porous structure volume fraction as a constraint condition;
s7: finite element analysis is carried out on the functional gradient porous structure, and node displacement and node strain energy density are solved;
s8: deriving objective functions and sensitivities of constraint conditions with respect to global design variables;
s9: with the aim of maximizing rigidity, updating global design variables by adopting a gradient-based optimization algorithm to obtain a porous structure with geometric and functional gradient distribution in space;
s10: judging whether the iterative process meets a convergence condition;
s11: and if the convergence is not met, repeating the steps S5-S10.
The mixed level set function Φ (x, t) of the cellular structure unit cell in said step S1 is defined as follows:
Figure GDA0002978540690000021
wherein the weight coefficient w is a variable related to time t, the value range is w is more than or equal to 0 and less than or equal to 1 (t), x is a space physical coordinate,
Figure GDA0002978540690000022
and
Figure GDA0002978540690000023
is a predefined cellular structure unit cell level set function.
In step S2, a plurality of mixed level set functions are defined on the design domain unit cell, each mixed level set function represents a cellular structure unit cell, and the definition of the cellular structure unit cell entity region Ω is implicitly described by the mixed level set function Φ (x, t) as follows:
Figure GDA0002978540690000031
where x is the spatial physical coordinate in the design domain D, phi (x, t)>The region of 0 represents the physical region in the design domain, Φ (x, t)<The area of 0 represents a hole area in the design domain,
Figure GDA0002978540690000038
is the boundary between the solid region and the void region.
In the step S3, the hamiltonian-jacobian partial differential equation of the control level set function evolution is converted into an ordinary differential equation, so as to realize time and space decoupling, and the mathematical expression is as follows:
Figure GDA0002978540690000032
in the formula (I), the compound is shown in the specification,
Figure GDA0002978540690000033
representing the derivation symbol, vnFor normal velocities of level set function boundary evolution,
Figure GDA0002978540690000034
modulo the gradient of the level set function.
The step S4 discretizes the design domain D to define a finite element mesh in the spatial coordinate system. An implicit geometric description method of a mixed level set function is adopted on a geometric model, and a finite element method is adopted on a physical model to calculate structural response.
The step S5 first defines a global design variable for each node, i.e., a set of global weighting coefficients, which may represent a set of vectors associated with time t:
w(t)＝[w1(t) w2(t) w3(t) … wm(t)]T
for any two-dimensional porous structure unit cell with the number of C, the weight coefficients on the four nodes are the sub-vectors of the global weight coefficient vector:
Figure GDA0002978540690000035
for any three-dimensional porous structure unit cell with the number of C, the weight coefficients on the eight nodes are the sub-vectors of the global weight coefficient vector:
Figure GDA0002978540690000036
where T is the transpose of a matrix or vector, SCTo select the matrix, the selection of the weight coefficients of the nodes of the porous structure cell with the number C from the global weight coefficient vector w (t) is effected.
For local design variables of unit cell internal nodes, namely weight coefficients of unit cell internal nodes, utilizationShape function NCAnd (6) carrying out interpolation.
The shape function of a two-dimensional quadrilateral unit cell is defined as follows:
Figure GDA0002978540690000037
the expansion is as follows:
Figure GDA0002978540690000041
in the formula, X, Y represents coordinates of a parent cell local coordinate system of a quadrangular cell.
The shape function of the three-dimensional hexahedral unit cell is defined as follows:
Figure GDA0002978540690000042
the expansion is as follows:
Figure GDA0002978540690000043
in the formula, X, Y, Z is the coordinates of the parent cell local coordinate system of the hexahedral cell.
Weight coefficient w on unit cell CC(t) is:
wC(x)＝NCw(t)＝NCSCw(t)
in the step S6, a functional gradient porous structure topology optimization mathematical model based on a mixed level set method is established with the global weight coefficient on the porous structure unit cell node as a design variable, the structural rigidity performance as an objective function, and the porous structure volume fraction as a constraint condition:
find:w(t)＝(w1(t) w2(t) w3(t) … wm(t))T
Figure GDA0002978540690000044
Figure GDA0002978540690000045
in the formula, m represents the number of global design variables, J represents the flexibility of the porous structure, epsilon represents the strain field of the structure, T represents the matrix transposition, u represents the displacement field of the structure, E represents the elastic matrix of the solid material, aФ(u,v)＝lФ(v) In a weak form of the elastic equilibrium equation, v represents a virtual displacement domain in the kinetically allowed displacement domain space U, g (Ω) is the volume constraint function of the structure, ζ represents the volume fraction of the structure,
Figure GDA0002978540690000051
representing the volume of the design field, wmaxAnd wminUpper and lower limits, w, of design variables, respectivelykRepresenting the global weight coefficients of the kth node in the design domain.
Step S7 is to carry out finite element solution on the structure and solve the state equation aФ(u,v)＝lФ(v) Obtaining node displacement, and energy bilinear form a in weak form of elastic equilibrium equationФ(u, v) and load linear form lФ(v) Respectively expressed as:
aΦ(u,v)＝∫ΩεT(u)Eε(v)dΩ
Figure GDA0002978540690000052
wherein f is applied at the boundary
Figure GDA0002978540690000053
The traction force on the part boundary of (1), p represents the volumetric force, and δ (Φ) is the derivative of Heaviside with respect to the Φ function.
The strain energy density of the quadrilateral unit node is obtained by unit strain energy expansion:
designing the strain energy density of the domain corner node: rhon＝ρ1e
Designing the strain energy density of a central node of a domain boundary:
Figure GDA0002978540690000054
design domain internal node strain energy density:
Figure GDA0002978540690000055
in the formula, ρ1e、ρ2e、ρ3e、ρ4eThe unit strain energy of 4 plane units adjacent to one node in the two-dimensional discrete design domain respectively.
The hexahedron unit node strain energy density is obtained by unit strain energy expansion:
designing the strain energy density of the domain corner node: rhon＝ρ1e
Designing the strain energy density of a central node of a domain boundary:
Figure GDA0002978540690000056
design domain surface internal node strain energy density:
Figure GDA0002978540690000057
design domain internal node strain energy density:
Figure GDA0002978540690000058
in the formula, ρ1e、ρ2e、ρ3e、ρ4e、ρ5e、ρ6e、ρ7e、ρ8eCell strain energy of 8 hexagonal cells adjacent to one node in a three-dimensional discrete design domain.
The sensitivity formula of the objective function with respect to the design variables in the step S8 is:
Figure GDA0002978540690000059
in the formula (I), the compound is shown in the specification,
Figure GDA0002978540690000061
for the sensitivity of the objective function, G (Φ) is the nodal strain energy density and δ (Φ) is the derivative of Heaviside with respect to the Φ function.
The sensitivity of the constraint function with respect to the design variables is given by the equation:
Figure GDA0002978540690000062
in the formula (I), the compound is shown in the specification,
Figure GDA0002978540690000063
for the sensitivity of the constraint, g (Ω) is the constraint.
The update strategy of the design variables in step S9 is as follows:
Figure GDA0002978540690000064
in the formula, n represents iteration step number, sigma is step length, max represents maximum value in brackets, min represents minimum value in brackets, eta is damping operator, and upper and lower limits of design variable are respectively designated as wmin＝0.001，w max1. B is an expression for sensitivity defined as follows:
Figure GDA0002978540690000065
in the formula, Λ is a Lagrange multiplier and is obtained by a dichotomy; the constant μ ═ 1 e-10.
The convergence criterion in step S10 is defined as follows:
Figure GDA0002978540690000066
or n is more than or equal to nmax
In the formula, theta represents a very small positive number, nmaxRepresenting the maximum number of iterations.
Generally, compared with the prior art, the above technical solution conceived by the present invention mainly has the following technical advantages:
1. the invention provides a mixed level set method for topological optimization of a functionally graded porous structure, which takes a global weight coefficient on a porous structure unit cell node as a design variable, converts an original Hamilton-Jacobian partial differential equation into a normal differential equation, can be combined with a mature and efficient optimization algorithm in the optimization field, reduces the complexity of solution and has a strict mathematical theory.
2. Meanwhile, the method reserves the advantages of the traditional level set method, can simultaneously carry out shape optimization and topology optimization, can clearly describe the topological structure and keeps the topological boundary smooth compared with a homogenization method and a density method.
3. In addition, the invention discloses a mixed level set method framework for topological optimization of a functional gradient porous structure, which can obtain the functional gradient porous structure with good connectivity and meeting the design requirements by predefining initial unit cell structures with different properties and has good expansibility.
Drawings
The invention is further illustrated by the following figures and examples.
FIG. 1 is a flow chart of a mixed level set method for topological optimization of a functionally graded porous structure provided by the invention.
FIG. 2 is a schematic diagram of the two-dimensional design domain discrete grid and structural topology optimization design variable definition of the present invention.
FIG. 3 is a schematic diagram of the three-dimensional design domain discrete grid and structural topology optimization design variable definition of the present invention.
FIG. 4 is a schematic diagram of 2 initial unit cell structures defined in the preferred embodiment of the present invention.
Figure 5 is a schematic diagram of a two-dimensional cantilever beam structure in a preferred embodiment of the invention.
Fig. 6 is a schematic diagram of the structure optimization result in the preferred embodiment of the present invention.
Detailed Description
Embodiments of the present invention will be further described with reference to the accompanying drawings.
Example 1:
referring to fig. 1, the present invention provides a mixed level set method for topological optimization of a functionally graded porous structure, as shown in fig. 1, the method comprising the steps of:
s1: based on two porous structure single cell configurations, an implicit level set function is adopted for geometric description to construct a single cell mixed level set function of a porous structure;
further, the mixing level set function Φ (x, t) of the porous-structure unit cell in the step S1 is defined as follows:
Figure GDA0002978540690000071
wherein the weight coefficient w is a variable related to time t, the value range is w is more than or equal to 0 and less than or equal to 1 (t), x is a space physical coordinate,
Figure GDA0002978540690000072
and
Figure GDA0002978540690000073
is a predefined cellular structure unit cell level set function.
S2: defining a plurality of mixed level set functions in a space coordinate system, wherein each mixed level set function represents a porous structure unit cell;
more specifically, in step S2, a plurality of mixed level set functions are defined on the design domain unit cell, each mixed level set function represents a cellular structure unit cell, and the mixed level set function Φ (x, t) is used to implicitly describe the definition of the cellular structure unit cell entity region Ω as follows:
Figure GDA0002978540690000081
where x is the spatial physical coordinate in the design domain D, phi (x, t)>The region of 0 represents the physical region in the design domain, Φ (x, t)<The area of 0 represents a hole area in the design domain,
Figure GDA0002978540690000082
is the boundary between the solid region and the void region.
S3: performing space-time decoupling on a Hamilton-Jacobian partial differential equation for controlling the evolution of the level set function, and parameterizing the mixed level set function;
more specifically, the original hamilton-jacobian equation is a partial differential equation, and the mathematical expression of the equation is as follows:
Figure GDA0002978540690000083
the mixed level set function disclosed by the invention is substituted into the original Hamilton-Jacobian partial differential equation to obtain an ordinary differential equation only related to time t, so that time and space decoupling is realized, and the mathematical expression of the equation is as follows:
Figure GDA0002978540690000084
in the formula (I), the compound is shown in the specification,
Figure GDA0002978540690000085
representing the derivation symbol, vnFor normal velocities of level set function boundary evolution,
Figure GDA0002978540690000086
modulo the gradient of the level set function.
S4: defining a finite element mesh in a space coordinate system, and matching an implicit geometric model with a physical model;
more specifically, the step S4 discretizes the design domain D, defining a finite element mesh in the spatial coordinate system. An implicit geometric description method of a mixed level set function is adopted on a geometric model, and a finite element method is adopted on a physical model to calculate structural response.
S5: as shown in fig. 2, macroscopic design variables are defined on four vertices of each planar porous structure unit cell, and microscopic design variables are defined inside the unit cell; as shown in fig. 3, macroscopic design variables are defined on eight vertices of each cubic porous structure unit cell, and microscopic design variables are defined inside the unit cell;
more specifically, first, a global design variable, i.e., a set of global weighting coefficients, is defined for each node, and the design variable may represent a set of vectors associated with time t:
w(t)＝[w1(t) w2(t) w3(t) … wm(t)]T
for any two-dimensional porous structure unit cell with the number of C, the weight coefficients on the four nodes are the sub-vectors of the global weight coefficient vector:
Figure GDA0002978540690000087
for any three-dimensional porous structure unit cell with the number of C, the weight coefficients on the eight nodes are the sub-vectors of the global weight coefficient vector:
Figure GDA0002978540690000091
where T is the transpose of a matrix or vector, SCTo select the matrix, the selection of the weight coefficients of the nodes of the porous structure cell with the number C from the global weight coefficient vector w (t) is effected.
For local design variables of the unit cell internal nodes, namely weight coefficients of the unit cell internal nodes, a shape function N is utilizedCAnd (6) carrying out interpolation.
The shape function of a two-dimensional quadrilateral unit cell is defined as follows:
Figure GDA0002978540690000092
the expansion is as follows:
Figure GDA0002978540690000093
in the formula, X, Y represents coordinates of a parent cell local coordinate system of a quadrangular cell.
The shape function of the three-dimensional hexahedral unit cell is defined as follows:
Figure GDA0002978540690000094
the expansion is as follows:
Figure GDA0002978540690000095
in the formula, X, Y, Z is the coordinates of the parent cell local coordinate system of the hexahedral cell.
Weight coefficient w on unit cell CC(t) is:
wC(x)＝NCw(t)＝NCSCw(t)
s6: establishing a functional gradient porous structure topological optimization mathematical model based on a mixed level set method by taking the structural rigidity performance as an objective function and the porous structure volume fraction as a constraint condition;
more specifically, a function gradient porous structure topological optimization mathematical model based on a mixed level set method is established by taking a global weight coefficient on a porous structure unit cell node as a design variable, taking the structure rigidity performance as an objective function and taking the porous structure volume fraction as a constraint condition:
find:w(t)＝(w1(t) w2(t) w3(t) … wm(t))T
Figure GDA0002978540690000101
Figure GDA0002978540690000102
wherein m represents the number of global design variables and J is a porous structureThe compliance of (a) is a strain field of the structure, T represents matrix transposition, u is a displacement field of the structure, E is an elastic matrix of the solid material, aФ(u,v)＝lФ(v) In a weak form of the elastic equilibrium equation, v represents a virtual displacement domain in the kinetically allowed displacement domain space U, g (Ω) is the volume constraint function of the structure, ζ represents the volume fraction of the structure,
Figure GDA0002978540690000103
representing the volume of the design field, wmaxAnd wminUpper and lower limits, w, of design variables, respectivelykRepresenting the global weight coefficients of the kth node in the design domain.
S7: finite element analysis is carried out on the functional gradient porous structure, and node displacement and node strain energy density are solved;
more specifically, the step S7 is to perform finite element solution on the structure, and solve the state equation aФ(u,v)＝lФ(v) Obtaining node displacement, and energy bilinear form a in weak form of elastic equilibrium equationФ(u, v) and load linear form lФ(v) Respectively expressed as:
aΦ(u,v)＝∫ΩεT(u)Eε(v)dΩ
Figure GDA0002978540690000104
wherein f is applied at the boundary
Figure GDA0002978540690000107
The traction force on the part boundary of (1), p represents the volumetric force, and δ (Φ) is the derivative of Heaviside with respect to the Φ function.
The strain energy density of the quadrilateral unit node is obtained by unit strain energy expansion:
designing the strain energy density of the domain corner node: rhon＝ρ1e
Designing the strain energy density of a central node of a domain boundary:
Figure GDA0002978540690000105
design domain internal node strain energy density:
Figure GDA0002978540690000106
in the formula, ρ1e、ρ2e、ρ3e、ρ4eThe unit strain energy of 4 plane units adjacent to one node in the two-dimensional discrete design domain respectively.
The hexahedron unit node strain energy density is obtained by unit strain energy expansion:
designing the strain energy density of the domain corner node: rhon＝ρ1e
Designing the strain energy density of a central node of a domain boundary:
Figure GDA0002978540690000111
design domain surface internal node strain energy density:
Figure GDA0002978540690000112
design domain internal node strain energy density:
Figure GDA0002978540690000113
in the formula, ρ1e、ρ2e、ρ3e、ρ4e、ρ5e、ρ6e、ρ7e、ρ8eCell strain energy of 8 hexagonal cells adjacent to one node in a three-dimensional discrete design domain.
S8: deriving objective functions and sensitivities of constraint conditions with respect to design variables;
more specifically, the sensitivity of the objective function with respect to the design variables in step S8 is derived in a formula as follows:
constructing a Lagrangian function:
Figure GDA0002978540690000114
derivative of the lagrangian function:
Figure GDA0002978540690000115
wherein, lambda is Lagrange multiplier, G (phi) is node strain energy density, delta (phi) is derivative of Heaviside with respect to phi function,
Figure GDA0002978540690000116
being the modulus of the gradient of the level set function, vnNormal velocity for level set function boundary evolution.
Obtaining the normal speed of level set function boundary evolution according to the decoupled Hamilton-Jacobian ordinary differential equation:
Figure GDA0002978540690000117
v is to benSubstituting the derivative of the lagrange function yields:
Figure GDA0002978540690000118
and then, the chain rule can obtain:
Figure GDA0002978540690000119
by comparing the above two formulas, sensitivity can be obtained
Figure GDA0002978540690000121
Figure GDA0002978540690000122
The sensitivity of the homologus constraint function with respect to design variables is given by the equation:
Figure GDA0002978540690000123
in the formula (I), the compound is shown in the specification,
Figure GDA0002978540690000124
for the sensitivity of the constraint, g (Φ) is the constraint.
S9: with the aim of maximizing rigidity, updating global design variables by adopting a gradient-based optimization algorithm to obtain a porous structure with geometric and functional gradient distribution in space;
Figure GDA0002978540690000125
in the formula, n represents iteration step number, sigma is step length, max represents maximum value in brackets, min represents minimum value in brackets, eta is damping operator, and upper and lower limits of design variable are respectively designated as wmin＝0.001，w max1. B is an expression for sensitivity defined as follows:
Figure GDA0002978540690000126
in the formula, Λ is a Lagrange multiplier and is obtained by a dichotomy; the constant μ ═ 1 e-10.
S10: judging whether the iterative process meets a convergence condition;
more specifically, the convergence criterion in step S10 is defined as follows:
Figure GDA0002978540690000127
or n is more than or equal to nmax
In the formula, theta represents a very small positive number, nmaxRepresenting the maximum number of iterations.
S11: and if the convergence is not satisfied, repeating the steps S5-S10.
Example 2:
fig. 4 shows a two-dimensional cantilever beam structure with a length L and a width W, where L is 2W, the left boundary freedom of the design domain is fully constrained, a vertically downward concentrated load is applied at the midpoint of the right boundary, the design domain is divided into 400 × 200 four-node planar quadrilateral units, and each unit cell size is defined as 40 × 40 four-node planar quadrilateral units according to a preferred embodiment of the present invention. FIG. 5 shows two kinds of porous unit cells defined in this example, which are subjected to structural topology optimization by the method provided by the present invention, and the result shown in FIG. 6 is obtained after convergence. The optimized result proves that the mixed level set method for topologically optimizing the functionally graded porous structure can obtain the porous structure with the geometric and functional gradient distribution in space, and the clear boundary and the good connectivity of the porous structure are ensured.

Claims (10)

1. A mixed level set method for topological optimization of a functionally graded porous structure is characterized by comprising the following steps:
s1: based on two porous structure single cell configurations, an implicit level set function is adopted for geometric description to construct a single cell mixed level set function of a porous structure;
s2: defining a plurality of mixed level set functions in a space coordinate system, wherein each mixed level set function represents a porous structure unit cell;
s3: performing space-time decoupling on a Hamilton-Jacobian partial differential equation for controlling the evolution of the level set function, and parameterizing the mixed level set function;
s4: defining a finite element mesh in a space coordinate system, and matching an implicit geometric model with a physical model;
s5: defining a global design variable on the vertex of each porous structure unit cell, and defining a local design variable inside the unit cell;
s6: establishing a functional gradient porous structure topological optimization mathematical model based on a mixed level set method by taking the structural rigidity performance as an objective function and the porous structure volume fraction as a constraint condition;
s7: finite element analysis is carried out on the functional gradient porous structure, and node displacement and node strain energy density are solved;
s8: deriving objective functions and sensitivities of constraint conditions with respect to global design variables;
s9: with the aim of maximizing rigidity, updating global design variables by adopting a gradient-based optimization algorithm to obtain a porous structure with geometric and functional gradient distribution in space;
s10: judging whether the iterative process meets a convergence condition;
s11: and if the convergence is not met, repeating the steps S5-S10.
2. The mixed level set method for topological optimization of a functionally graded porous structure according to claim 1, wherein: the mixed level set function Φ (x, t) of the cellular structure unit cell in said step S1 is defined as follows:
Figure FDA0002978540680000011
wherein the weight coefficient w is a variable related to time t, the value range is w is more than or equal to 0 and less than or equal to 1 (t), x is a space physical coordinate,
Figure FDA0002978540680000012
and
Figure FDA0002978540680000013
is a predefined cellular structure unit cell level set function.
3. The mixed level set method for topological optimization of a functionally graded porous structure according to claim 1, wherein: in step S2, a plurality of mixed level set functions are defined on the design domain unit cell, each mixed level set function represents a cellular structure unit cell, and the definition of the cellular structure unit cell entity region Ω is implicitly described by the mixed level set function Φ (x, t) as follows:
Figure FDA0002978540680000021
where x is the spatial physical coordinate in the design domain D, phi (x, t)>The region of 0 represents the physical region in the design domain, Φ (x, t)<The area of 0 represents a hole area in the design domain,
Figure FDA0002978540680000022
is the boundary between the solid region and the void region.
4. The mixed level set method for topological optimization of a functionally graded porous structure according to claim 1, wherein: in the step S3, the hamiltonian-jacobian partial differential equation of the control level set function evolution is converted into an ordinary differential equation, so as to realize time and space decoupling, and the mathematical expression is as follows:
Figure FDA0002978540680000023
in the formula (I), the compound is shown in the specification,
Figure FDA0002978540680000024
representing the derivation symbol, vnFor normal velocities of level set function boundary evolution,
Figure FDA0002978540680000025
modulo the gradient of the level set function.
5. The mixed level set method for topological optimization of a functionally graded porous structure according to claim 1, wherein: the step S4 discretizes the design domain D, defines a finite element mesh in the spatial coordinate system, adopts an implicit geometric description method of a hybrid level set function on the geometric model, and calculates a structural response by a finite element method on the physical model.
6. The mixed level set method for topological optimization of a functionally graded porous structure according to claim 1, wherein: the step S5 first defines a global design variable for each node, i.e., a set of global weighting coefficients, which may represent a set of vectors associated with time t:
w(t)＝[w1(t) w2(t) w3(t) … wm(t)]T
for any two-dimensional porous structure unit cell with the number of C, the weight coefficients on the four nodes are the sub-vectors of the global weight coefficient vector:
Figure FDA0002978540680000026
for any three-dimensional porous structure unit cell with the number of C, the weight coefficients on the eight nodes are the sub-vectors of the global weight coefficient vector:
Figure FDA0002978540680000027
where m denotes the number of global design variables, T is the transpose of a matrix or vector, SCSelecting the weight coefficient of each node of the porous structure unit cell with the number of C from the global weight coefficient vector w (t) for selecting the matrix;
for local design variables of the unit cell internal nodes, namely weight coefficients of the unit cell internal nodes, a shape function N is utilizedCCarrying out interpolation;
the shape function of a two-dimensional quadrilateral unit cell is defined as follows:
Figure FDA0002978540680000031
the expansion is as follows:
Figure FDA0002978540680000032
wherein X, Y is the local coordinate system coordinate of the mother unit of the quadrilateral unit;
the shape function of the three-dimensional hexahedral unit cell is defined as follows:
Figure FDA0002978540680000033
the expansion is as follows:
Figure FDA0002978540680000034
wherein X, Y, Z is the parent unit local coordinate system coordinate of the hexahedral unit;
weight coefficient w on unit cell CC(t) is:
wC(x)＝NCw(t)＝NCSCw(t) 。
7. the mixed level set method for topological optimization of a functionally graded porous structure according to claim 1, wherein: in the step S6, a functional gradient porous structure topology optimization mathematical model based on a mixed level set method is established with the global weight coefficient on the porous structure unit cell node as a design variable, the structural rigidity performance as an objective function, and the porous structure volume fraction as a constraint condition:
find:w(t)＝(w1(t) w2(t) w3(t) … wm(t))T
min:
Figure FDA0002978540680000041
s.t:
Figure FDA0002978540680000042
in the formula, m represents the number of global design variables, J represents the flexibility of the porous structure, epsilon represents the strain field of the structure, T represents the matrix transposition, u represents the displacement field of the structure, E represents the elastic matrix of the solid material, aФ(u,v)＝lФ(v) In a weak form of the elastic equilibrium equation, v represents a virtual displacement domain in the kinetically allowed displacement domain space U, g (Ω) is the volume constraint function of the structure, ζ represents the volume fraction of the structure,
Figure FDA0002978540680000043
representing the volume of the design field, wmaxAnd wminUpper and lower limits, w, of design variables, respectivelykRepresenting the global weight coefficients of the kth node in the design domain.
8. The mixed level set method for topological optimization of a functionally graded porous structure according to claim 1, wherein: step S7 is to carry out finite element solution on the structure and solve the state equation aФ(u,v)＝l Ф(v) And obtaining node displacement, calculating according to the displacement and the unit stiffness matrix to obtain unit strain energy density, and expanding the unit strain energy density into the node strain energy density.
9. The mixed level set method for topological optimization of a functionally graded porous structure according to claim 1, wherein: the sensitivity formula of the objective function with respect to the design variables in the step S8 is:
Figure FDA0002978540680000044
in the formula (I), the compound is shown in the specification,
Figure FDA0002978540680000045
g (phi) is the node strain energy density, and delta (phi) is the derivative of Heaviside with respect to the phi function;
the sensitivity of the constraint function with respect to the design variables is given by the equation:
Figure FDA0002978540680000046
in the formula (I), the compound is shown in the specification,
Figure FDA0002978540680000047
for the sensitivity of the constraint, g (Ω) is the constraint.
10. The mixed level set method for topological optimization of a functionally graded porous structure according to claim 1, wherein: the update strategy of the design variables in step S9 is as follows:
Figure FDA0002978540680000051
in the formula, n represents iteration step number, sigma is step length, max represents maximum value in brackets, min represents minimum value in brackets, eta is damping operator, and upper and lower limits of design variable are respectively designated as wmin＝0.001，wmaxB is an expression for sensitivity defined as follows:
Figure FDA0002978540680000052
in the formula, Λ is a Lagrange multiplier and is obtained by a dichotomy; constant μ ═ 1 e-10;
the convergence criterion in step S10 is defined as follows:
Figure FDA0002978540680000053
or n is more than or equal to nmax
In the formula, theta represents a very small positive number, nmaxRepresenting the maximum number of iterations.
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Citations (6)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
US20110270587A1 (en) * 2008-09-11 2011-11-03 Kyoto University Structural optimization system, structural optimization methodlogy, and structural optimization program
CN108897956A (en) * 2018-07-02 2018-11-27 清华大学 A kind of porous mechanical Parts optimization design method
US20190026415A1 (en) * 2016-11-04 2019-01-24 South University Of Science And Technology Of China Product structure design method
CN109670207A (en) * 2018-11-22 2019-04-23 华中科技大学 A kind of dynamics integrated design method towards a variety of porous material structures
CN111859693A (en) * 2020-07-28 2020-10-30 大连理工大学 Efficient porous structure representation and optimization method
CN111985137A (en) * 2020-08-21 2020-11-24 华中科技大学 Multivariable cutting level set optimization method for porous structure topology optimization

Patent Citations (6)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
US20110270587A1 (en) * 2008-09-11 2011-11-03 Kyoto University Structural optimization system, structural optimization methodlogy, and structural optimization program
US20190026415A1 (en) * 2016-11-04 2019-01-24 South University Of Science And Technology Of China Product structure design method
CN108897956A (en) * 2018-07-02 2018-11-27 清华大学 A kind of porous mechanical Parts optimization design method
CN109670207A (en) * 2018-11-22 2019-04-23 华中科技大学 A kind of dynamics integrated design method towards a variety of porous material structures
CN111859693A (en) * 2020-07-28 2020-10-30 大连理工大学 Efficient porous structure representation and optimization method
CN111985137A (en) * 2020-08-21 2020-11-24 华中科技大学 Multivariable cutting level set optimization method for porous structure topology optimization

Non-Patent Citations (2)

* Cited by examiner, † Cited by third party
Title
具有吸能和承载特性的多孔结构拓扑优化;杜义贤 等;《华中科技大学学报（自然科学版）》;20190831;第47卷(第8期);第108-113页 *
基于能量均匀化的高剪切强度周期性点阵结构拓扑优化;杜义贤 等;《机械工程学报》;20170930;第53卷(第18期);第152-160页 *

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