CN106844917A - A kind of lathe base method of topological optimization design based on support reaction Variance Constraints - Google Patents

A kind of lathe base method of topological optimization design based on support reaction Variance Constraints Download PDF

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CN106844917A
CN106844917A CN201710016069.7A CN201710016069A CN106844917A CN 106844917 A CN106844917 A CN 106844917A CN 201710016069 A CN201710016069 A CN 201710016069A CN 106844917 A CN106844917 A CN 106844917A
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support reaction
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高彤
邱利彬
赵志光
张卫红
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Northwestern Polytechnical University
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Abstract

The invention discloses a kind of lathe base method of topological optimization design based on support reaction Variance Constraints.It uses the degree average to weigh support reaction distribution of the support reaction variance size in the specified free degree of node at fixed support.With lathe base entirety compliance as optimization aim during optimization, the design that topologizes is carried out as constraint using specified support reaction variance and structural volume.The optimization configuration for obtaining effectively improves its lathe base and fixes the situation of support reaction skewness at support, so as to strengthen stability of the lathe in real work, makes lathe remain to keep preferable precision in permanent work.Lathe base method of topological optimization design increases support reaction Variance Constraints in topological optimization model, the stability requirement of lathe work is taken into full account while construction weight is mitigated, design obtains brand-new clear effective node configuration, realize the Creative conception design of lathe base, design efficiency is high, more possesses practical implementation demand.

Description

A kind of lathe base method of topological optimization design based on support reaction Variance Constraints
Technical field
It is more particularly to a kind of to be based at fixed support node the present invention relates to a kind of lathe base method of topological optimization design The lathe base method of topological optimization design of support reaction Variance Constraints.
Background technology
Lathe base is the most important support member of lathe, is the whole basis of lathe.Its physical dimension and distribution form, Determine the dynamic characteristic of itself.Therefore, lathe base structure design is reasonably carried out, is expected while base weight is mitigated Reduce base deformation and vibration at work, the performance that whole machine is lifted with this is just seemed and is even more important.
Document " based on Optistruct chain digital control gear hobbing machine lathe bed topology optimization design " (《Machine-building》, 2011) in A kind of bed piece construction design method is disclosed, based on traditional Topology Optimization Method, the topological optimization for realizing bed piece sets Meter, realizes the light-weight design of structure on the premise of target requirement is reached.
Document " based on finite element analysis technology VK50 numerical control lathe bed milling machine tools base design " (《Lathe and hydraulic pressure》, 2008 " it is related to a kind of construction design method of lathe base in.Using finite element analysis software to lathe base mould in document Type carries out static and dynamic analysis and simulation calculation, by analytical calculation find original structure design defect, then to it is original design into Row is targetedly improved, and calculating is analyzed again to the structure after improvement, by this step cycle, finally obtains more satisfied knot Really.
In above-mentioned document, stability and the required precision of lathe work cannot be considered when grinding machine structure design is carried out, and The stability of understructure can be influenceed in lathe real work due to long-term jitter accumulation, causes the machining accuracy of lathe to obtain not To guarantee.Therefore consider that the stability of lathe work is extremely necessary with required precision in the lathe base design phase.
The content of the invention
In order to avoid the deficiency that prior art is present, the present invention proposes a kind of lathe base based on support reaction Variance Constraints Method of topological optimization design.Be incorporated into topological optimization technology in lathe base structure design by the method, with lathe base entirety The minimum optimization aim of compliance, using structural volume as constraint；The method is weighed at fixed support with support reaction variance yields Node support reaction average degree, the support reaction variance conduct for adding lathe base to fix at support node in topological optimization model Constraints carries out topology optimization design to structure, ensures lathe stability at work with this.
The inventive method solves the technical scheme that its technical problem used:A kind of machine based on support reaction Variance Constraints Bed base method of topological optimization design, it is characterised in that comprise the following steps:
Step 1. sets up FEM model；FEM meshing is carried out to lathe base structure initial geometric model, is obtained FEM model is obtained, topology design variable x is setiInitial value, its value changes between 0-1 in optimization, and i is positive integer, represents Design domain element number, 1≤i≤ne, neDesign domain inner structure unit sum is represented, to avoid stiffness matrix unusual, topology is set The lower limit x of design variableL；
Step 2. sets load and boundary condition to lathe base structural finite element model；According to actual condition, not Appropriate simplified is made to load and boundary condition on the premise of influence global analysis, load and perimeter strip are carried out in finite element software The applying of part；
Step 3. gives solid material Young's modulus E(0)With Poisson's ratio μ；After each iteration, according to current design variate-value, The respective material attribute in structural finite element model is updated, in the method, is calculated respectively often using SIMP materials interpolation model Young's modulus E of one finite elements under current iteration stepi
The Young's modulus of subscript (0) presentation-entity material, penalty factor p value takes 3；
Step 4. carries out finite element analysis to the analysis model that three above step is set up, and obtains structural stiffness matrix, knot Support reaction vector information at structure motion vector, fixed support；
For linear static analysis, structure finite element equilibrium equation can be written as
KU=F (2)
In formula, K is structure Bulk stiffness matrix, and U is modal displacement vector, and F is panel load vector；
For ease of trying to achieve the counter-force size at fixed support node as, above equilibrium equation is write the form of matrix in block form
The free degree at subscript c representative structure non-supported, subscript behalf structure fixes the free degree at support；Correspondingly, UcGeneration Modal displacement vector, U at table non-supportedsRepresent modal displacement vector, F at fixed supportaRepresent structure overall load at non-supported Vector, R represents node support reaction vector at fixed support；
It is 0, i.e. U to fix displacement at support according to boundary conditions=0, equation (3) can abbreviation be
Equation (4) is launched by row respectively, such as formula (5), you can in the hope of Uc, then by can be calculated fixed support Locate node support reaction vector R
Step 5. calculates lathe understructure entirety compliance
C=UTKU (6)
Lathe understructure entirety compliance C is calculated according to formula (6)；It is by extracting finite element analysis knot during specific calculating The stiffness matrix k of i-th unit in fruitiWith motion vector ui, the compliance of each unit is first calculated using self-compiling programThe compliance of all units is stacked up again obtains the overall compliance C of structure；
Step 6. calculates lathe understructure and fixes node support reaction variance at support
Specify the support reaction variance size in the free degree flat come the distribution of quantitative description support reaction using node at fixed support Equal index；Support reaction variance D (R) can be calculated as the following formula
Wherein nrRepresent the support reaction total number for considering, ΩRVRepresent the free degree set for considering that support reaction is average, RjRepresent J-th support reaction value in the free degree set for being considered, j represents ΩRVIn each support reaction numbering, 1≤j≤nr；Table Show the average value of the support reaction of all considerations, it is calculated as follows
Obviously, support reaction variance D (R) is always on the occasion of smaller expression of its value specifies the support reaction distribution in the free degree more flat ；
Step 7. given volume constraint upper limit VU, the structural volume V that current iteration is walked is calculated using formula (9)C,
Wherein ViRepresent i-th volume of solid material unit；
Step 8. defines topological optimization model
It is as follows using the topology optimization design model with support reaction Variance Constraints
X represents the set of design variable, x in formulaiIt is i-th design variable value of finite elements, presentation-entity material Whether there is, wherein 1≤i≤ne；To avoid unusual, the introducing design variable lower limit x of structural stiffness matrix during finite element analysisL, neGeneration Table design domain unit sum, optimization aim is minimum for the overall compliance C of structure；In constraints comprising volume and variance about Beam, VURepresent design domain volume constraint VCThe upper limit, ViRepresent i-th volume of solid material unit in design domain；
Step 9. carries out sensitivity analysis and Optimized Iterative
The sensitivity of object function and constraints on design variable is tried to achieve, is optimized repeatedly using topological optimization program In generation, obtain the lathe base topology optimization design result based on support reaction Variance Constraints.
Beneficial effect
A kind of lathe base method of topological optimization design based on support reaction Variance Constraints proposed by the present invention, using fixation Node is specified the support reaction variance size in the free degree and is distributed average degree weighing support reaction at support.With lathe during optimization Base entirety compliance is optimization aim, and the design that topologizes is carried out as constraint using specified support reaction variance and structural volume. The optimization configuration for finally giving can be effectively improved the situation that lathe base fixes support reaction skewness at support, so as to strengthen machine Stability of the bed in real work, it is ensured that precision of the lathe in permanent work.In lathe base method of topological optimization design In, the support reaction variance upper limit constrained in the corresponding free degree of all fixed support nodes is 500, and iteration is optimized to embodiment Clearly design result is can obtain after 300 steps.Support reaction maximum is at node at its fixation support of structure after optimization 100.59N, average value is 28N, and support reaction variance is 498.6, effectively improves lathe base and fixes support reaction distribution at support Uneven situation.Lathe base method of topological optimization design is solving to fix the corresponding free degree of node at support with lathe base On Variance Constraints Structural Topology Optimization Design problem when be highly effective.Lathe base method of topological optimization design is being opened up Flutterring in Optimized model increases meeting counter-force Variance Constraints, and the stability of lathe work has been taken into full account while construction weight is mitigated It is required that, design obtains brand-new clear effective node configuration, realizes the Creative conception design of lathe base；Design efficiency Height, possesses engineering background, more conforms to engineering actual demand.
Brief description of the drawings
It is excellent to a kind of lathe base topology based on support reaction Variance Constraints of the present invention with implementation method below in conjunction with the accompanying drawings Change method for designing to be described in further detail.
Fig. 1 is lathe base method of topological optimization design flow chart of the present invention based on support reaction Variance Constraints.
Fig. 2 is that the initial geometric model load of method for designing of the present invention applies schematic diagram.
Fig. 3 is that the initial geometric model boundary condition of method for designing of the present invention applies schematic diagram.
Fig. 4 is lathe base design effect figure in method for designing application example of the present invention.
Specific embodiment
The present embodiment is a kind of lathe base method of topological optimization design based on support reaction Variance Constraints.
Refering to Fig. 1~4, the lathe base method of topological optimization design that this exemplary application is based on support reaction Variance Constraints is directed to Certain lathe base is designed analysis, comprises the following steps that:
A () sets up lathe base geometrical model using geometric modeling software UG, model is tamped with reference to engineering design After operation, obtain carrying out the initial geometric model of topology optimization design, its basic size is 1.427m long, width 0.75m, height 0.47m.FEM meshing is carried out to the geometrical model using the pre-processing module Hypermesh of HyperWorks softwares. Carry out GTD model first, and carry out the division of design domain and non-design domain, during mesh generation using based on hexahedron, pentahedron Supplemented by grid cell, cell type is Solid45；Construction unit sum is 120558, and wherein design domain unit sum is 98608, non-design domain unit sum is 21950；After the completion of FEM model is set up in Hypermesh, derivation can be in Ansys The cdb FEM model files of middle reading；The topology design variable x of design domain unitiIt is just that initial value is disposed as 0.35, i Integer, represents design domain element number, 1≤i≤ne, neRepresent structure design domain unit sum.Additionally, to ensure optimum results Symmetrically, the design variable to design domain unit in Ansys carries out symmetrical treatment；To avoid stiffness matrix unusual, topology is set The lower limit x of design variableL=10-3。
B () sets load and boundary condition；Base top subjects the pressure loading that lathe other parts are brought, wherein The pressure size that " returning " shape boss bears is 193KPa, and both sides strip guide rail bears pressure size for 35.3KPa, middle small side Platform bears pressure size for 6.3KPa；6 zonules of base bottom obtain fixed support by foundation bolt；It is final to realize fixing Support reaction at support tends to average, and the purpose of the more permanent steady operation of lathe is allowed to reach.
(c) given solid material Young's modulus E(0)=200GPa, Poisson's ratio μ=0.3；After each iteration, according to currently setting Meter variate-value, updates the respective material attribute in structural finite element model；Calculate each respectively using SIMP materials interpolation model Young's modulus E of the individual finite elements under current iteration stepi
The Young's modulus of subscript (0) presentation-entity material, penalty factor p value takes 3.
D () utilizes general finite element analysis Ansys, finite element fraction is carried out to the analysis model that three above step is established Analysis, acquiring unit stiffness matrix, element displacement vector and support reaction vector at fixed support,
For linear static analysis, structure finite element equilibrium equation can be written as
KU=F (2)
In formula, K is structure Bulk stiffness matrix, and U is modal displacement vector, and F is panel load vector.
For ease of trying to achieve the counter-force size at stationary nodes as, above equilibrium equation is write the form of matrix in block form
The free degree at subscript c representative structure non-supported, subscript behalf structure fixes the free degree at support；Correspondingly, UcGeneration Modal displacement vector, U at table non-supportedsRepresent modal displacement vector at fixed support；FaRepresent structure overall load at non-supported Vector, R represents node support reaction vector at fixed support.
It is 0, i.e. U to fix displacement at support according to boundary conditions=0, equation (3) can abbreviation be
Equation (4) is launched by row respectively, such as formula (5), you can in the hope of Uc, saved at fixed support by be can be calculated Point support reaction vector R
E () calculates lathe understructure entirety compliance
C=UTKU (6)
Lathe understructure entirety compliance C is calculated according to formula (6)；By extracting Finite element analysis results during specific calculating In i-th unit stiffness matrix kiWith motion vector ui, the compliance of each unit is first calculated using self-compiling programThe compliance of all units is stacked up again obtains the overall compliance C of structure.
F () calculates lathe understructure and fixes node support reaction variance at support
Specify the support reaction variance size in the free degree flat come the distribution of quantitative description support reaction using node at fixed support Equal index；Support reaction variance D (R) is calculated as the following formula
Wherein nrRepresent the support reaction total number for considering, ΩRVRepresent the free degree set for considering that support reaction is average, RjRepresent J-th support reaction value in the free degree set for being considered, j represents ΩRVIn each support reaction numbering, 1≤j≤nr；Table Show the average value of the support reaction of all considerations, it is calculated as follows
It will be apparent that support reaction variance D (R) is always on the occasion of smaller expression of its value specifies the support reaction distribution in the free degree to get over Averagely.
G () calculates the structural volume V of current iteration step using formula (9)C
Wherein ViRepresent i-th volume of solid material unit.
Total quality upper limit 1500Kg is given in the present embodiment, design domain volume upper limit V is thus calculatedU。
H () defines topological optimization model
It is as follows using the topology optimization design model with support reaction Variance Constraints
X represents the set of design variable, x in formulaiIt is i-th design variable value of finite elements, presentation-entity material Whether there is, be worth for 1 when represent at this as solid material, be worth for 0 when represent at this without material.All design variable initial values are respectively provided with It is 0.35.To avoid unusual, the introducing design variable lower limit x of structural stiffness matrix during finite element analysisL=10-3, neRepresentative sets Meter domain unit sum is 98608.Optimization aim is minimum for the overall compliance C of structure.Volume and variance are included in constraints Constraint；VURepresent design domain volume constraint VCThe upper limit, ViRepresent i-th volume of solid material unit in design domain.
Additionally, the upper limit of node support reaction variance D (R) of all considerations is DU；Generally, when design is optimized, DUBy Determined using the support reaction variance yields of the topology optimization design result for not considering Variance Constraints.In the present embodiment, in Variance Constraints Limit DU=500.
I model is carried out a finite element analysis by () first in each Optimized Iterative；Then by optimizing sensitivity point Analysis, tries to achieve the sensitivity of object function and constraints on design variable.Optimized repeatedly using self-editing topological optimization program In generation, obtain the lathe base topology optimization design result based on support reaction Variance Constraints.In the present embodiment, based on the excellent of gradient Changing algorithm such as ConLin, GCM, MDPA, SLP, QP can realize Optimized Iterative.
In the present embodiment, the support reaction variance upper limit constrained in the corresponding free degree of all fixed support nodes is 500, to reality Apply after example optimizes the step of iteration 300 and can obtain clearly design result.Design structure configuration clear and rational, the total matter of understructure Amount is no more than 1500Kg, and its quality mitigates significantly.Structure support reaction variance is 498.6 optimize simultaneously after, meets given constraint.This Example shows the validity of the lathe base Structural Topology Optimization Design method based on support reaction Variance Constraints.Considering support reaction After Variance Constraints, optimum results node support reaction maximum at fixed support is reduced to 100.6N, and average value is 28N, and branch is anti- Power variance is reduced to 498.6.Illustrate the optimum results for obtaining branch at its fixation support in the case of guarantee base integral rigidity Counter-force is average, so as to improve the stability of lathe work, makes lathe remain to keep preferable precision in permanent work.

Claims (1)

1. a kind of lathe base method of topological optimization design based on support reaction Variance Constraints, it is characterised in that including following step Suddenly:
Step 1. sets up FEM model；FEM meshing is carried out to lathe base structure initial geometric model, is had Limit meta-model, sets topology design variable xiInitial value, its value changes between 0-1 in optimization, and i is positive integer, represents design Domain element number, 1≤i≤ne, neDesign domain inner structure unit sum is represented, to avoid stiffness matrix unusual, topology design is set The lower limit x of variableL；
Step 2. sets load and boundary condition to lathe base structural finite element model；According to actual condition, do not influenceing Load and boundary condition made on the premise of global analysis it is appropriate simplify, load and boundary condition are carried out in finite element software Apply；
Step 3. gives solid material Young's modulus E(0)With Poisson's ratio μ；After each iteration, according to current design variate-value, update Respective material attribute in structural finite element model, in the method, each is calculated using SIMP material interpolation models respectively Young's modulus E of the finite elements under current iteration stepi
E i = x i p E ( 0 ) - - - ( 1 )
The Young's modulus of subscript (0) presentation-entity material, penalty factor p value takes 3；
Step 4. carries out finite element analysis to the analysis model that three above step is set up, and obtains structural stiffness matrix, structure position Support reaction vector information at the amount of shifting to, fixed support；
For linear static analysis, structure finite element equilibrium equation can be written as
KU=F (2)
In formula, K is structure Bulk stiffness matrix, and U is modal displacement vector, and F is panel load vector；
For ease of trying to achieve the counter-force size at fixed support node as, above equilibrium equation is write the form of matrix in block form
K c c K c s K c s T K s s U c U s = F a R - - - ( 3 )
The free degree at subscript c representative structure non-supported, subscript behalf structure fixes the free degree at support；Correspondingly, UcRepresent non- Modal displacement vector, U at supportsRepresent modal displacement vector, F at fixed supportaRepresent at non-supported structure overall load to Amount, R represents node support reaction vector at fixed support；
It is 0, i.e. U to fix displacement at support according to boundary conditions=0, equation (3) can abbreviation be
K c c K c s K c s T K s s U c 0 = F a R - - - ( 4 )
Equation (4) is launched by row respectively, such as formula (5), you can in the hope of Uc, then by can be calculated section at fixed support Point support reaction vector R
K cc U c = F a K cs T U c = R - - - ( 5 )
Step 5. calculates lathe understructure entirety compliance
C=UTKU (6)
Lathe understructure entirety compliance C is calculated according to formula (6)；It is specific be when calculating by extracting Finite element analysis results in I-th unit stiffness matrix kiWith motion vector ui, the compliance of each unit is first calculated using self-compiling programThe compliance of all units is stacked up again obtains the overall compliance C of structure；
Step 6. calculates lathe understructure and fixes node support reaction variance at support
Specify the support reaction variance size in the free degree to be distributed come quantitative description support reaction using node at fixed support averagely to refer to Mark；Support reaction variance D (R) can be calculated as the following formula
D ( R ) = 1 n r Σ j ( R j - R ‾ ) 2 , j ∈ Ω R V - - - ( 7 )
Wherein nrRepresent the support reaction total number for considering, ΩRVRepresent the free degree set for considering that support reaction is average, RjExpression is examined J-th support reaction value in the free degree set of worry, j represents ΩRVIn each support reaction numbering, 1≤j≤nr；Represent institute There is the average value of the support reaction of consideration, it is calculated as follows
R ‾ = 1 n r Σ j R j , j ∈ Ω R V - - - ( 8 )
Obviously, support reaction variance D (R) is always on the occasion of smaller expression of its value specifies the support reaction distribution in the free degree average；
Step 7. given volume constraint upper limit VU, the structural volume V that current iteration is walked is calculated using formula (9)C,
V C = Σ i V i x i - - - ( 9 )
Wherein ViRepresent i-th volume of solid material unit；
Step 8. defines topological optimization model
It is as follows using the topology optimization design model with support reaction Variance Constraints
f i n d x = { x i } , i = 1 , 2 , ... , n e min C = U T K U s . t K U = F D ( R ) &le; D U V C = &Sigma; i V i x i &le; V U 0 < x L &le; x i &le; 1 - - - ( 10 )
X represents the set of design variable, x in formulaiIt is i-th design variable value of finite elements, the presence or absence of presentation-entity material, Wherein 1≤i≤ne；To avoid unusual, the introducing design variable lower limit x of structural stiffness matrix during finite element analysisL, neRepresentative sets Meter domain unit sum, optimization aim is minimum for the overall compliance C of structure；Volume and Variance Constraints, V are included in constraintsU Represent design domain volume constraint VCThe upper limit, ViRepresent i-th volume of solid material unit in design domain；
Step 9. carries out sensitivity analysis and Optimized Iterative
The sensitivity of object function and constraints on design variable is tried to achieve, iteration is optimized using topological optimization program, Obtain the lathe base topology optimization design result based on support reaction Variance Constraints.
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