AU2020103808A4 - A design method of the fail-safe topology optimization of continuum structures with the frequency and displacement constraints - Google Patents
A design method of the fail-safe topology optimization of continuum structures with the frequency and displacement constraints Download PDFInfo
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Abstract
This invention discloses a design method of the fail-safe topology optimization with the
frequency and displacement constraints on continuum structures. Firstly, a finite element model
is established based on the basic structure. The number of failure cases is determined, and the
5 initial center coordinate and other parameters for the failure region are entered. The optimization
parameters of displacement and frequency constraints are entered to form an optimization model.
The static analysis of the fail-safe structure including failure cases is carried out in turn, and the
explicit expression of the displacement constraint is established. The modal analysis of the fail
safe structure is carried out, and explicit expression of the frequency constraint is established.
0 The fail-safe optimization model established with constraints of the displacement and frequency
is expressed approximately as a mathematical formula and solved. The displacement and
fundamental frequency of each failure case, and the optimized fail-safe configuration, can be
obtained after inversion. This invention provides a reference to the fail-safe topology
optimization problem of the field of statics combined with dynamics.
5
1/4
Start
Establish the finite element model
Input optimization parameters
Conduct structural analysis by MSC.Nastran<
Exrc aayi rslsModify elastic modulus and
Extrct aalyss reultsupdate the finite element
model
Form optimization parameters and calculate
NO
Converge?
Ye s
Inversion
End
Figure I
Description
1/4
Start
Establish the finite element model
Input optimization parameters
Conduct structural analysis by MSC.Nastran<
Exrc aayi rslsModify elastic modulus and Extrct reultsupdate a lys the finite element model
Form optimization parameters and calculate
NO Converge?
Ye s
Inversion
End
Figure I
[0001] The present application claims priority from Chinese Patent Application No. 202010050300.6 titled "A design method of the fail-safe topology optimization with the frequency and displacement constraints on continuum structures" and filed on 17 January 2020, the content of which is hereby incorporated by reference in its entirety.
[0002] The present disclosure relates to the technical field of engineering structure design and particularly involves a design method of the fail-safe topology optimization of continuum structures with the frequency and displacement constraints.
[0003] Topology optimization is to find the optimal load path under the given loading and boundary conditions, while optimizing the material distribution. With the rapid development of finite element theory and computer software technology, this optimization method is gradually developed. Compared with size optimization and shape optimization, structural topology optimization can determine more design parameters, save materials more significantly, and achieve greater economic benefits. At the same time, it can provide engineering designers with a conceptual design in the initial stage of the structural design.
[0004] The traditional topology structure obtained by continuum topology optimization is usually the optimal load path under certain external loads and takes full advantage of the material performance. However, this optimal configuration tends to the statically determinate structure owning the disadvantage of lacking redundancy. In this case, local structure failure can easily lead to overall failure. In fields with high safety requirements such as aerospace, ships, high-speed trains, bridges, and ultra-high buildings, the high sensitivity of topological structures to local failure will make it more economical but less safe. In the engineering field with high safety requirements, it is often regarded as economical but unsafe. Therefore, we should pay more attention to the fail-safe topology optimization.
[0005] It is necessary to consider the stiffness and vibration of engineering structures. In practical engineering applications, people will require the structure to meet certain stiffness conditions which can realize by confining the displacement at certain nodes within the allowable range, to guarantee that the structural displacement and deformation do not exceed the prescribed constraint value. This is the optimization problem with displacement constraints. At the same time, the dynamic response of structures relates to both its natural frequency and the frequency of the dynamic load. In extreme cases, when the frequency of the dynamic load is very close to the structural natural frequency, the structure will resonate. Therefore, the structure's natural frequency needs to be restricted in the structural dynamic design. For example, the aircraft design of rockets and missiles has strict requirements on the structure's natural frequency. Therefore, confining displacement and frequency has great significance to the topology optimization design for continuum structures.
[0006] There is thus a need to provide improved design methods for the fail-safe topology optimization of continuum structures with the frequency and displacement constraints or to at least provide a useful alternative to existing methods.
[0007] Aiming at the traditional topology optimization design problem of continuum structures with displacement and frequency constraints, this invention considers the fail-safe design conception, and proposes a design method of the fail-safe topology optimization of continuum structures with the frequency and displacement constraints. Combined with the technique for the finite element static and modal analysis, this design method considering that the local failure exists in the internal structure can effectively solve the problem that traditional structural components in the displacement and frequency optimization are too "efficient". This design method can consider both stiffness and natural frequency characteristics of the optimal structure, reduce the sensitivity of the overall stiffness and fundamental frequency of the optimal structure to the local failure, and improve the redundancy of the optimal structure. This has important theoretical significance and extensive engineering application value for structural static and dynamic optimization in many fields such as aviation, aerospace, automobiles, bridges, and civil engineering.
[0008] Embodiments of this invention considers the fail-safe design conception, adopts the placement strategy that local failure occupies the structural domain evenly without gap and overlap, and combines the fail-safe design conception with the topology optimization of continuum structures with frequency and displacement constraints for the first time. This fail-safe topology optimization method can greatly improve the redundancy of the optimal structure, and significantly reduce the sensitivity of the overall stiffness and frequency of the optimal structure to local failure. It has strong theoretical significance and application value in the field of structural topology optimization. To achieve the above purpose, embodiments of this invention may adopt the following technical scheme.
[0009] According to a first aspect, there is provided a design method of the fail-safe topology optimization of continuum structures with the frequency and displacement constraints including the following steps:
[0010] The first step is to determine the basic structure, and build up a finite element model.
[0011] The second step is to determine the number of failure cases, and input the initial center coordinate and other parameters for the failure region.
[0012] The third step is to enter optimization parameters of displacement and frequency constraints to form a fail-safe topology optimization model with displacement and frequency constraints.
[0013] The fourth step is to carry out the static analysis of the fail-safe structure containing failure cases and establish the explicit expression of the displacement constraint.
[0014] The fifth step is to conduct the modal analysis of the fail-safe structure and establish the explicit expression of the frequency constraint.
[0015] The sixth step is to form the mathematical fail-safe optimization formula with the constraint of the displacement and frequency, and it can be solved by mathematical programming algorithms.
[0016] The seventh step is to realize the inversion of topological variables to obtain the optimized topological configuration and the displacement magnitude and natural frequency of the corresponding structural failure case.
[0017] Embodiments of the invention may provide advantages over the existing techniques. The results of traditional structural topology optimization with displacement and frequency constraints are similar to discrete truss structures, in which the traditional result will be overall failure when some members occur the material failure deriving from collision, explosion, or corrosion. Compared with the traditional structural topology optimization design with displacement and frequency constraints, this invention fully considers the structural fail-safe design conception, adds damage cases to the optimization design, and the optimal fail-safe result can still satisfy the frequency and displacement constraints after occurring local failure, namely, it can normally carry external loads but not occur overall failure.
[0018] Embodiments of the present disclosure will be discussed with reference to the accompanying drawings wherein:
[0019] Figure 1 is a flow chart of the fail-safe topology optimization design with continuum displacement and frequency constraints;
[0020] Figure 2 is the basic structure and optimal topology results;
[0021] Figure 3 is iterative process diagrams of the fail-safe topology optimization;
[0022] Figure 4 is an iterative curve graph where (a) shows the iterative curve of the total volume and (b) plots the iterative curve of the fundamental frequency for each failure case.
[0023] In the following description, like reference characters designate like or corresponding parts throughout the figures.
[0024] As shown in Fig.1, this invention provides a design method of the fail-safe topology optimization of continuum structures with the frequency and displacement constraints, including the following concrete solution:
[0025] The first step is to determine the basic structure, and build up a finite element model:
[0026] First of all, it needs to determine the maximum design boundary to optimize the continuous structure based on the design requirement, namely determine the basic structure, also define the failure and the non-failure design domain based on the performance requirement. Secondly, based on MSC.Patran, the geometric model of the basic structure is established and the finite element mesh is divided. Thirdly, it needs to define materials, give element attributes, apply boundary conditions, set up two different load cases of static analysis and modal analysis. Finally, the output options of the static analysis and modal analysis are selected.
[0027] The second step is to determine the number of failure cases, and input the initial center coordinate and other parameters for the failure region:
[0028] In the main program written by the PCL (Patran Command Language) which is the built-in programming language of the MSC.Patran software, the number of failure cases determined according to the practical problem should be set at first. Then it needs to enter the two-dimensional center coordinate of the initial failure region. Finally, some parameters for the failure region should be inputted to determine the initial square failure region: the square of the half of the diagonal line and the square of the half of the side length.
[0029] The third step is to enter optimization parameters of displacement and frequency constraints to form a fail-safe topology optimization model with displacement and frequency constraints:
[0030] Firstly, according to the performance and design requirements of the structural rigidity and natural frequency, we need to input optimization parameters of the displacement and frequency constraint including the numerical magnitude, direction, and node number which the external force acts on the displacement constraint, also including the numerical magnitude of the fundamental frequency constraint, convergence accuracy, and filter radius. Then the size and shape of the failure region, and the displacement and frequency constraint can be determined. A fail-safe optimization model with displacement and frequency constraints is formulated as follows:
Find tE EN N Make W Y f,(t)w ->min
0:! t' 1 (i = 1,.., N: I =1,.., L)
where t is the element topology design variable vector. t, is the topology variable of element i, and 1/0
indicates that the element is existent or non-existent. t E EN represents that the element topology design
variable vector t belongs to the vector in the n-dimensional Euclidean space. W is the total weight, and
wi is the inherent weight of the i-th element. u, is the displacement of the l-th failure case and u is the
corresponding displacement constraint. f() f, (t), f,(t) are the corresponding filter functions of
the weight, stiffness, and mass mentioned on theICM(Independent, Continuous, Mapping) method. A, is
the first eigenvalue of the l-th failure case, and Ai is the given first eigenvalue constraint of the l-th
failure case. L means the total number of failure cases and N represents the total number of elements.
[0031] The fourth step is to carry out the static analysis of the fail-safe structure containing failure cases and establish the explicit expression of the displacement constraint:
[0032] The basic structure containing a local failure region is called a structural failure case. In the PCL main program, starting from the initial failure region, the local failure is implemented sequentially on the initial structure according to the number of failure cases, and each structural failure case needs to conduct
static analysis. The vector of nodal forces of the i-th element under the external load F' is extracted from each structural failure case in turn. The displacement vector of the i-th element under a unit force can be extracted from the corresponding unit virtual load case, then the contribution coefficient of the i-th element in the l-th failure case S, =(61.)T can be obtained. So this explicit expression between the N displacement and the design variable can be formulated as u,(x)= ( V )T TR
[0033] The fifth step is to conduct the modal analysis of the fail-safe structure and establish the explicit expression of the frequency constraint:
[0034] The modal analysis result (U,, and V, )can be extracted from each structural failure case by turns.
U= -- uT ku, is the strain energy of element i corresponding to the fundamental frequency of the l-th
failure case. 1J,-= 2uTku, is the kinetic energy of element i corresponding to the fundamental 2 frequency of the l-th failure case. Thus, the derivative of the first eigenvalue of each failure case to the design variables namely sensitivity analysis results can be calculated by the corresponding strain energy and kinetic energy which can be extracted from finite element modal analysis of each failure case. And these results provide mechanical performance parameters of structural elements for the establishment of explicit constraint equations.
[0035] The sixth step is to form the mathematical fail-safe optimization formula with the constraint of the displacement and frequency, and it can be solved by mathematical programming algorithms:
[0036] Let the design variableX 1/ , and * is the power index of the stiffness filter function,
taking lk=5 here. Based on the sensitivity analysis and Taylor expansion, the optimization model is explicitly processed, and the corresponding explicit equation of the quadratic programming can be obtained as follows:
Find xE EN 0O W=N Make W=$ -min
s.t. cix, < b,
Ax : d, (l1,.. L) B
ci=DxS(tj(')) /u rIc,=DxS(t())" /u,
B,=ef ,"fk"'(t,)/na;Si,=(oV)TTR b, =D; d, =, /Z 1: x, : x . (i= 1,.,N). (2)
where a=y),//y , y,=2 and Y*=5 are the power index of the weight filter function and the stiffness
filter function, respectively. c and bI are the constant coefficient of the explicit displacement constraint
inequality. du and e are the constant coefficient of the explicit frequency constraint inequality. The
superscript V represents the iterative times. S is the displacement contribution coefficient of the i-th
element in the l-th failure case. 6 is the nodal displacement vector of the i-th element under unit virtual
FR load cases in the l-th failure case, and is the nodal force vector of the i-th element under real load
AMv) U!v) Vv U A , U/u> case. ' is the difference between U' andii . WhenU 1 /or- /,it takes D =1; when
or -lit takes D = -1. XI is the upper limit of topological variables.
[0037] To make topological variables approach 0 or 1 as near as possible, it needs to integrate the discrete condition of topological variables into the primitive objective function of minimal weight based on the linear weighted method to get a new single objective function. Then the second-order Taylor expansion is applied to the above new objective function, and the approximate expression can be obtained as
ax+b x -> min (3)
[0038] So we can update the approximate mathematical optimization formula as follows:
Find xE EN
Make W=Za,x,+bIx -> min
a(a+2) a(a+2) 2a(2a+2) a, =-1 x x, x,
+ a(a+1) a(a+1) 2a(2a+1) a(a+1) 2a(2a+1)
s.t. cx C'X b,
d,,x, : e,(l=1,...,L)
c, =Dx S,(t r/uic =Dx S,(t) u,
d =-1xDx Ad,=1xDx A 0/ kA x, - x,0)4 N __N_ b, =D; e,=Dx A -A4(x')2A A e,=Dx A,-A. (x()-Z2A '/IA
[0039] Considering that the number of design variables in the optimization model (4) far exceeds the number of constraints, it can be transformed into a dual model based on the Kuhn-Tucker conditions and then solved by the sequential quadratic programming algorithm. If the solution satisfies the convergence accuracy, output the optimal solution and proceed to step 7; if not, then modify the topological variables, and return to the fourth step to perform the next finite element analysis and optimization model processing until the convergence accuracy condition is reached.
[0040] The seventh step is to realize the inversion of topological variables in order to obtain the optimized topological configuration and the displacement magnitude and natural frequency of the corresponding structural failure case:
[0041] According to the cloud chart of the topological optimization results, appropriate delete threshold selected for the topological variables can make continuous topological variables mapped inversely to 0-1 discrete topology variables, and obtain the optimized topological configuration and the displacement magnitude and natural frequency of each structural failure case
[0042] This invention discloses a design method of the fail-safe topology optimization with the frequency and displacement constraints on continuum structures, including the following steps: (1) Determine the basic structure, and build up a finite element model. (2) Determine the number of failure cases, and input the initial center coordinate and other parameters for the failure region. (3) Enter optimization parameters of displacement and frequency constraints to form a fail-safe topology optimization model with displacement and frequency constraints. (4) Carry out the static analysis of the fail-safe structure containing failure cases, and establish the explicit expression of the displacement constraint. (5) Conduct the modal analysis of the fail-safe structure, and establish the explicit expression of the frequency constraint. (6) Form the mathematical fail-safe optimization formula with the constraint of the displacement and frequency, and it can be solved by mathematical programming algorithms. (7) Realize the inversion of topological variables in order to obtain the optimized topological configuration and the displacement magnitude and natural frequency of the corresponding structural failure case. This invention can effectively solve the fail-safe topology optimization design problem with the continuum displacement and frequency constraints. Compared with the traditional topology optimization with displacement and frequency constraints, this invention can significantly improve the redundancy of the optimal result and provides a reference to the fail-safe topology optimization problem of the field of statics combined with dynamic.
[0043] EXAMPLE
[0044] The specific implementation steps of this invention are described in detail with a fail-safe topology optimization example considering displacement and frequency constraints.
[0045] As shown in Fig. 2(a), the first step is to establish the basic structure by the MSC.Patran. The basic structure is a rectangular plane plate with dimensions of 200 mm x 100 mm x 9 mm, meshed into x 20 = 800 rectangular elements. The left boundary of this structure is fixed, and the concentrated load at the midpoint of the right boundary is P--15600N. The material of this basic structure has a Young's modulus E = 68890MPa, Poisson's ratio p= 0.3, density p=1.0x10 9 Mg/mm3 . The fundamental frequency of the basic structure is 1247 Hz, and the frequency constraint is that the fundamental frequency of each failure case is not less than 800 Hz. The finite element model is showed in Fig. 2(b). Considering eight failure cases illustrated in Fig. 2(c) which are evenly distributed in the basic structure, each failure region is a square with side length d = 50 mm. Finally, it needs to set the static and modal analysis case, and select to output the element nodal force and element strain energy in the corresponding output requests.
[0046] The second step is to set parameters for the failure case in the main program written by the PCL. Firstly, the number of failure cases is set as 8. Then it needs to enter the two-dimensional center coordinate of the initial failure region (25,25). Finally, some parameters for the failure region should be inputted to determine the initial square failure region including the square of the half of the diagonal line with the value 225and the square of the half of the side length with the value 225.
[0047] The third step is to input optimization parameters of the displacement and frequency constraint. Displacement optimization parameters include: displacement constraint in the y direction, the sign of constraint inequality >, the constraint value -0.6mm, and the constrained node number 441. Frequency optimization parameters include: the sign of constraint inequality >, the constraint value 800Hz, the convergence accuracy 0.001, and the filtering radius 7.5. Then the size and shape of the failure region, and the displacement and frequency constraint can be determined
[0048] The fourth step is to conduct static analysis on the initial structural failure case by using MSC.Nastran. The static analysis and modal analysis of each structural failure case are performed in turn by the FOR loop. Automatic extraction of the element nodal force from the static analysis case realizing by the program provides mechanical performance parameters of structural elements for the establishment of explicit equations of the displacement constraint.
[0049] The fifth step is to conduct modal analysis on the initial structural failure case. Automatic extraction of the element kinetic energy and strain energy from the modal analysis case realizing by the program provides mechanical performance parameters of structural elements for the establishment of explicit equations of the displacement constraint.
[0050] The sixth step is to form the mathematical fail-safe optimization formula with the constraint of the displacement and frequency based on the sensitivity analysis and Taylor expansion, and it can be solved by mathematical programming algorithms. If the solution satisfies the given convergence accuracy, output the optimal solution; if not, then modify the topological variables and update the finite element model, and return to the fourth step to perform the next finite element analysis and optimization model processing until the convergence accuracy condition is reached.
[0051] The seventh step is to realize the inversion of topological variables to obtain the optimized topological configuration
[0052] The optimized result obtained by the fail-safe topology optimization with displacement and frequency constraints is illustrated in Fig. 2(e). When the inversion threshold is 0.6724, the optimized structure after inversion is shown in Fig. 2(f). Table 1 and Table 2 list some values for the optimized fail safe structure. Compared with the traditional optimized topological configuration shown in Fig. 2(d), the redundancy of the optimized fail-safe structure can be significantly improved, and the sensitivity of the optimized fail-safe structure to local failure can be greatly reduced. It can be seen from the iterative history of the fail-safe topology optimization in Figure 3 that the optimal path to carry the load and bear the frequency response is preserved clearly. At the same time, it can be seen from the iteration curve of structural volume in Fig. 4 that the total volume of the structure considering fail-safe design converges stably to 66292mm 3 after 38 iterations. The iterative curve of the fundamental frequency for each failure case shown in Fig. 4 converges stably. As listed in Table 2, it can be concluded that most of the structural failure cases can meet the displacement constraint. The structural displacement is -6.36mm when the 4th and 6th region occur failure. This value slightly exceeds the constraint value by 0.36mm, which can be considered to satisfy the requirement of displacement constraint. Therefore, the method proposed in this
I1
invention can realize the fail-safe topology optimization design of continuum structures with displacement and frequency constraints.
TABLE 1 Optimized fail-safe topological results of the frequency case.
Volume/mm3 Fundamental frequency Fundamental frequency Iteration number before inversion/Hz after inversion /Hz
38 66292 896 929
TABLE2 Displacement of optimized structure for various failure cases.
Failure Case 1 2 3 4 5 6 7 8
Displacement -5.48 -5.40 -4.57 -6.36 -4.57 -6.36 -5.48 -5.40
[0053] It will be understood that the terms "comprise" and "include" and any of their derivatives (eg comprises, comprising, includes, including) as used in this specification is to be taken to be inclusive of features to which the term refers, and is not meant to exclude the presence of any additional features unless otherwise stated or implied
[0054] The reference to any prior art in this specification is not, and should not be taken as, an acknowledgement of any form of suggestion that such prior art forms part of the common general knowledge.
[0055] It will be appreciated by those skilled in the art that the disclosure is not restricted in its use to the particular application or applications described. Neither is the present disclosure restricted in its preferred embodiment with regard to the particular elements and/or features described or depicted herein. It will be appreciated that the disclosure is not limited to the embodiment or embodiments disclosed, but is capable of numerous rearrangements, modifications and substitutions without departing from the scope as set forth and defined by the following claims.
Claims (7)
1. A design method of the fail-safe topology optimization with the frequency and displacement constraints on continuum structures has the characteristics of the following steps: The first step is to determine the failure and the non-failure design domain of the continuum structure, build up a finite element model which is the same as the general finite element modeling process, and establish the element meshes, element properties, element materials, load cases, and output results; The second step is to input the number of failure cases, the initial center coordinate, and other parameters for the failure region into the finite element model, so that the location and shape of the initial failure region are completely determined; The third step is to input optimization parameters of the displacement and frequency constraint, including the numerical magnitude, direction, and node number which the external force acts on the displacement constraint, also including the numerical magnitude of the fundamental frequency constraint, convergence accuracy, and filter radius; then a fail-safe optimization model with displacement and frequency constraints is established; In the fourth step, the basic structure with a local failure region is called a structural failure case. Starting from the initial failure region, the initial structure implements local failure sequentially according to the number of failure cases. Each structural failure case corresponds to a static load case which can perform static analysis to build up the corresponding virtual load case, and the explicit expression of displacement constraint can be obtained through the adjoint method of the displacement sensitivity analysis; The fifth step is to carry out the modal analysis in each structural failure case, and the explicit expression of frequency constraints can be obtained by extracting the modal analysis result of each structural failure case; The sixth step is to form the mathematical fail-safe optimization formula with the objective of minimal weight and the constraint of the displacement and frequency, and it can be solved by mathematical programming algorithms; and The seventh step is to realize the inversion of topological Variables in order to obtain the optimized topological configuration and the displacement magnitude and natural frequency of the corresponding structural failure case.
2. The design method as claimed in claim 1, wherein the first step comprises: First of all, it needs to determine the maximum design boundary to optimize the continuous structure, also define the failure and the non-failure design domain based on the design requirements. Secondly, based on MSC.Patran, the geometric model of the basic structure is established and the finite element mesh is divided. Thirdly, it needs to define materials, give element attributes, apply boundary conditions, set up two different load cases of static analysis and modal analysis. Finally, the output options of the static analysis and modal analysis are selected.
3. The design method as claimed in claim 2, wherein the second step comprises: In the main program written by the PCL which is the built-in programming language of the MSC.Patran software, the number of failure cases determined according to the practical problem should be set at first. Then it needs to enter the two-dimensional center coordinate of the initial failure region. Finally, some parameters for the failure region should be inputted to determine the initial square failure region: the square of the half of the diagonal line and the square of the half of the side length.
4. The design method as claimed in claim 3, wherein the third step comprises: Firstly, according to the performance and design requirements of the structural rigidity and natural frequency, we need to input optimization parameters of the displacement and frequency constraint including the numerical magnitude, direction, and node number which the external force acts on the displacement constraint, also including the numerical magnitude of the fundamental frequency constraint, convergence accuracy, and filter radius. Then the size and shape of the failure region, and the displacement and frequency constraint can be determined. A fail-safe optimization model with displacement and frequency constraints is formulated as follows:
N
W = f(t,), -> min
s.t. u1(fk(ti)) ul (1
0 < t,I 1
(i = 1,.., N: I =,.., L)
where t is the element topology design variable vector. t, is the topology variable of element i,
and 1/0 indicates that the element is existent or non-existent. t E EN represents that the element topology design variable vector t belongs to the vector in the n-dimensional Euclidean space. W is the total weight, and w, is the inherent weight of the i-th element. u, is the displacement of thel-th failure case
and u is the corresponding displacement constraint. f,,(t,), f, (), f,,(t) are the corresponding filter
functions of the weight, stiffness, and mass mentioned on the ICM method. A, is the first eigenvalue of
the l-th failure case, and i is the given first eigenvalue constraint of the l-th failure case. L means the total number of failure cases and N represents the total number of elements.
5. The design method as claimed in claim 4, wherein the fourth step comprises: In the PCL main program, starting from the initial failure region, the local failure is implemented sequentially on the initial structure according to the number of failure cases, and each structural failure case needs to conduct static analysis. The vector of nodal forces of the i-th element under the external
load FR is extracted from each structural failure case in turn. The displacement vector of the i-th element
under a unit force can be extracted from the corresponding unit virtual load case, then the contribution
coefficient of the i-th element in the l-th failure case S (v)TFR can be obtained. So this explicit
expression between the displacement and the design variable can be formulated as N ui(x= _ (ov)F*
6. The design method as claimed in claim 5, wherein the fifth step comprises: Starting from the initial failure region, the local failure is implemented sequentially on the initial structure according to the number of failure cases, and the modal analysis need to be performed on each
structural failure case. The modal analysis result (U,, and V,, ) can be extracted from each structural failure
case by turns. U,=- Iu Tk u, is the strain energy of element i corresponding to the fundamental frequency 2
of the l-th failure case. 1 I- IIuTkiu,is the kinetic energy of element i corresponding to the 2 fundamental frequency of the l-th failure case. Thus, the derivative of the first eigenvalue of each failure case to the design variables namely sensitivity analysis results can be calculated by the corresponding strain energy and kinetic energy which can be extracted from finite element modal analysis of each failure case. And these results provide mechanical performance parameters of structural elements for the establishment of explicit constraint equations.
7. The design method as claimed in claim 6, wherein the sixth step comprises:
Let the design variables, 1/t1 , and 7* is the power index of the stiffness filter function,
taking 7k=5 here. Based on the sensitivity analysis and Taylor expansion, the optimization model is explicitly processed, and the corresponding explicit equation of the quadratic programming can be obtained as follows:
Find x E EN
Make W=$ -).min i=1 X,
s.t. cA b,
t(),/U 2 Ai DxSI( ' 2 ,kU -i d=-1xDx-2A" A/Ad =1 2-A,)A
N N
b, =D; e, =Dx ,- A,(x'))-I 2A,1 RMe, =D x A- 2(x(')- 2A, v) Ik
S, =(AT R (v) = U ,") -V ("); 1:! x, 5 (i1.,N). (2)
where a=y /Yk y =2 and 7*=5 are the power index of the weight filter function and the
stiffness filter function, respectively. cl and bi are the constant coefficient of the explicit displacement
constraint inequality. dl and el are the constant coefficient of the explicit frequency constraint inequality.
The superscript V represents the iterative times. S is the displacement contribution coefficient of the i-th
element in the l-th failure case. i is the nodal displacement vector of the i-th element under unit virtual
FR load cases in the l-th failure case, and is the nodal force vector of the i-th element under real load
AC')~~U UUV'u u U/ case. 4v)is the difference between " and V . WhenUI U1 or/" I ,it takes D =1; when
or - , it takes D = - '.i is the upper limit of topological variables; To make topological variables approach 0 or 1 as near as possible, it needs to integrate the discrete condition of topological variables into the primitive objective function of minimal weight based on the linear weighted method to get a new single objective function; then the second-order Taylor expansion is applied to the above new objective function, and the approximate expression can be obtained as
N
aix, + b ~x*min (3)
So we can update the approximate mathematical optimization formula as follows:
Find xeEN
Make W=Eax,+b,x7 ->min
, )a(a+-2) a(a+2) 2a(2a+2) X_____ x, 2-1 -1 X a+___ X x x, a(a+1) a(a+1) 2a(2a+1) a(a+1) 2a(2a+1) X a2 X -+ 2-2 -2 X2-2 x, x, x, x, x N s.t. ca,&b,
dY x e, (I 1.. L )
c=D x S,(t ) /Ic,= D x S,(t )/U, 2 2 d = -1 x D x -A4 / AR du -1 xD x - A(," /,
b, =D; e, =DxL*Z - Zk(X )Z il/* -k I L Ax(kZA~
S ( )F ()= /," V|';1:! x,. i 1.. N).
Considering that the number of design variables in the optimization model (4) far exceeds the number of constraints, it can be transformed into a dual model based on the Kuhn-Tucker conditions and then solved by the sequential quadratic programming algorithm.
Applications Claiming Priority (2)
| Application Number | Priority Date | Filing Date | Title |
|---|---|---|---|
| CN202010050300.6 | 2020-01-17 | ||
| CN202010050300.6A CN111241738A (en) | 2020-01-17 | 2020-01-17 | Continuum displacement and frequency constraint topology optimization design method considering damage-safety condition |
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