CN112966410B - Additive manufacturing self-supporting structure topology optimization method suitable for variable critical angle - Google Patents
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Abstract
The invention discloses a topological optimization method of an additive manufacturing self-supporting structure, which is suitable for variable critical angles, and comprises the following steps: 1. pretreatment: defining a structural geometric model, dispersing a design domain, adding boundary conditions, initializing design variables and setting optimization parameters; 2. topological optimization of the self-supporting structure: constructing a topological optimization mathematical model of the additive manufacturing self-supporting structure with variable critical angle and solving to obtain the optimal layout of the unit density of the design domain; 3. and (3) post-treatment: and setting the unit size, and extracting a density isosurface to obtain an optimal self-supporting structure design scheme. The invention adopts a design domain of a four-node rectangular unit discrete structure with a variable length-width ratio, constructs a self-supporting unit filtering method with a variable critical angle based on the design domain, establishes a topological optimization method of an additive manufacturing self-supporting structure with the variable critical angle, optimally designs the self-supporting structure with the variable critical angle, realizes the unsupported printing process, and saves printing materials and time.
Description
Technical Field
The invention belongs to the related field of structural optimization design, and particularly relates to a topological optimization method for an additive manufacturing self-supporting structure with a variable critical angle.
Background
In recent years, the additive manufacturing technology has been rapidly developed due to its outstanding advantage of being able to manufacture complex structures, providing a powerful tool for structure preparation, and widening the space of structure design. However, additive manufacturing has some process constraints, especially for the overhanging features, support is often required to be added below the overhanging features during forming, and the overhanging features are manually removed after printing is completed, which greatly wastes printing material and time, and also causes damage to the performance and surface quality of the parts.
In view of this problem, some researchers have studied a topological optimization method of a self-supporting structure for additive manufacturing, which aims to solve the problem of adding support from the structural design stage, but at present, only specific critical angles are restricted, and the restriction conditions of different critical angles are not considered.
Disclosure of Invention
The invention aims to provide a topological optimization method of an additive manufacturing self-supporting structure, which is suitable for variable critical angles, realizes the optimal design of the self-supporting structure with variable critical angles, saves printing materials and time, and solves the problem that the traditional topological optimization structure needs to be supported in the printing process.
The invention is realized by adopting the following technical scheme:
a method for topological optimization of an additive manufacturing self-supporting structure adapted to variable critical angles, comprising the steps of:
1) Pretreatment: defining a structural geometric model, dispersing a design domain, adding boundary conditions, initializing design variables and setting optimization parameters;
2) Topological optimization of the self-supporting structure: constructing a topological optimization mathematical model of the additive manufacturing self-supporting structure, which is suitable for changing the critical angle, and solving to obtain the optimal layout of the unit density of the design domain;
3) And (3) post-treatment: and setting the unit size, and extracting a density isosurface to obtain an optimal design scheme of the self-supporting structure.
The invention is further improved in that the specific steps of the step 1) are as follows:
101 Defining a geometric model of the structure;
102 Setting a critical angle alpha, defining the length and width of the four-node rectangular unit as a and b respectively, and satisfying b = a × tan alpha;
103 Adopting nelx × nely four-node rectangular unit discrete design domain, if nelx and nely are both integers, then the design domain discrete design is completed, otherwise, rounding is carried out on nelx and nely, and the length and width of the unit are respectively adjusted to be a 'and b', adopting the adjusted unit discrete design domain, and the adjusted critical angle
104 Adding boundary conditions, applying load F, setting penalty factor P, elastic modulus E, poisson ratio mu and filtering radius r min Volume constraint f and convergence condition optimization parameters;
105 ) initializes the cell design variable x.
The further improvement of the invention is that in the step 2), based on the discretization design domain, a filtration method of the self-supporting unit with variable critical angle is constructed, and the filtration and the density filtration of the self-supporting unit are sequentially carried out, so that the topological optimization mathematical model of the additive manufacturing self-supporting structure with variable critical angle is constructed by taking the minimized flexibility, namely the maximized rigidity, as an objective function and taking the structural volume fraction as a constraint condition.
A further improvement of the invention consists in optimizing the mathematical model as follows:
an objective function: minimizing compliance
Constraint conditions are as follows: v (x) P (x))≤f.V 0 ；
Designing variables: x is more than or equal to 0 and less than or equal to 1;
wherein c is the structural flexibility, V is the structural volume, V 0 For the design domain total volume, f is the specified volume fraction, x is the design variable vector, x P Is a physical density vector.
The invention has the further improvement that the specific steps of the step 2) are as follows:
201 Self-supporting unit filtration to obtain unit self-supporting density, the expression is:
wherein x is ij Designing variables for the cells;
is a cell free standing density;
for the self-supporting judging factor, the expression is
Wherein β is an approximation parameter; t is a threshold value; s ij For the average density of the support unit set, the support unit set contains target units x ij Three adjacent cells of the lower layer
The number of boundary units is two, i and j respectively represent the positions of the units in the horizontal direction and the vertical direction;
202 Density filtering to obtain the physical density of the unit, the expression is:
wherein
Is the unit physical density;
is a cell free standing density; n is a radical of hydrogen e Is a unit set in the filtering area; h ei Is a weight coefficient expressed as H ei ＝max(0,r min - Δ (e, i)), wherein r min For the filter radius, Δ (e, i) is the distance between the centers of the cells i and e;
203 Finite element analysis calculates the structure flexibility and the structure volume, and the flexibility expression is as follows:
the volume expression is as follows:
wherein U is an overall displacement matrix; k is an integral rigidity matrix; n is the number of the discretization design domain units; u. of e Is a cell displacement matrix; k is a radical of e In the form of a matrix of cell stiffness,
is the cell physical density;
204 Sensitivity analysis, the sensitivity expression of the objective function to the design variable is:
the sensitivity expression of the constraint on the design variables is:
where c is the objective function, V is the volume constraint, x is the cell design variable, x F Is a unit self-supporting density, x P Is the unit physical density;
205 Solving the updating unit design variable x by adopting a moving asymptote method;
206 Calculating the maximum difference value of the updated design variable and the design variable of the previous step, judging whether the maximum difference value is smaller than the convergence value, if so, terminating the loop iteration, otherwise, repeating the steps.
The further improvement of the invention is that in the step 3), the size of the four-node rectangular unit is set, and the density isosurface is extracted based on the optimal layout of the unit density in the design domain, so as to obtain the optimal design scheme of the self-supporting structure.
The invention has at least the following beneficial technical effects:
the method can set any variable self-supporting critical angle, defines the size of a four-node rectangular unit based on the critical angle, adopts the discrete design domain of the unit, obtains the physical density of the unit through self-supporting unit filtration and density filtration in sequence, performs finite element analysis and sensitivity analysis, updates the design variable of the unit, iterates circularly until convergence, finally obtains the optimal unit density distribution, extracts the unit density isosurface and obtains the optimal self-supporting structure design scheme. Therefore, the method can optimally design the self-supporting structure for different self-supporting critical angles of different additive manufacturing processes, and the optimal structure does not need to be additionally supported during additive manufacturing;
further, the method comprises the steps of setting any critical angle alpha, defining the length and the width of a four-node rectangular unit based on the critical angle, enabling the unit width ratio to be equal to the tangent value of the critical angle, adopting the unit to discretely design a domain, if the number of the units is not an integer, rounding, and finely adjusting the critical angle until the number of the units is an integer, so that the random variable critical angle can be realized;
furthermore, the constructed minimized flexibility topological optimization mathematical model can meet the condition of structure volume constraint to minimize the structure flexibility and realize the optimal design of the self-supporting structure;
furthermore, the invention provides a self-supporting unit filtering method based on the Heaviside function, the method judges the self-supporting property of the unit through the Heaviside function, the product of the unit design variable and the self-supporting judgment factor can effectively reserve the self-supporting unit, and deletes the non-self-supporting unit, so that only the self-supporting unit is reserved in the design domain, and finally the self-supporting density of the unit is obtained; the chessboard pattern solution phenomenon can be effectively eliminated by adopting density filtering, and the unit physical density is finally obtained by solving the unit self-supporting density weighted average value of the density filtering area; finite element analysis is carried out based on unit physical density, an initial stiffness matrix of a four-node rectangular unit is calculated, the unit stiffness matrix is solved through a SIMP model, an overall stiffness matrix is obtained through assembly, a stiffness equation is solved to obtain an overall displacement matrix, structural flexibility and structural volume are calculated, sensitivity analysis is carried out, an updated design variable is solved by adopting a moving asymptote method, and optimal unit density distribution can be obtained through cyclic iteration until convergence;
furthermore, the visualization of the optimal self-supporting structure can be realized by setting the corresponding unit size and extracting the unit physical density isosurface.
In summary, the invention adopts a four-node rectangular unit discrete structure design domain with a variable length-width ratio, and establishes a self-supporting unit filtering method with a variable critical angle based on the method, so as to establish a topological optimization method for the additive manufacturing self-supporting structure with a variable critical angle, and the optimization design is adapted to the self-supporting structure with a variable critical angle, thereby realizing the unsupported printing process and saving the printing materials and time.
Drawings
FIG. 1 is a flow chart of the present invention;
FIG. 2 is a schematic diagram of a cantilever design domain;
FIG. 3 is a schematic diagram of a variable aspect ratio four-node rectangular cell;
fig. 4 is a schematic diagram of the discretized design domain and the self-supporting cell decision, wherein fig. 4 (a) shows a critical angle of 30 °, fig. 4 (b) shows a critical angle of 45 °, and fig. 4 (c) shows a critical angle of 60 °.
Detailed Description
The invention is further described below with reference to the following figures and examples.
The method proposed by the present invention can be used for the planar problem of the design of the self-supporting structure facing the additive manufacturing, and the following describes the implementation process of the present invention with reference to the case and the accompanying drawings, and fig. 1 shows the flowchart of the present invention.
(1) Pretreatment
The method comprises the following specific steps:
1) Defining a geometrical model of the cantilever beam, optimally designing the cantilever beam structure along the + Y direction as shown in fig. 2, and performing calculation and explanation by taking L =160 and h =100 as an example;
2) Setting a critical angle α, defining the length and width of a four-node rectangular unit as a and b respectively, and satisfying b = a × tan α, as shown in fig. 3, taking α =30 °, a =1, b = tan 30 ° as an example for calculation and explanation;
3) Calculating by adopting nelx multiplied by nely four-node rectangular unit discrete design domain
Wherein, nely is not integer, need to round nely, nely is approximately equal to 277, and the width of adjusting unit
The length-width ratio is 1
A schematic diagram of discretized design domains of different critical angles, as shown in FIG. 4;
4) Adding boundary conditions that the bottom of a design domain is fixed, a unit load along the + X direction is applied to the midpoint of the top, and setting a penalty factor P to be 3, the elastic modulus E =1 and poiseApparent ratio mu =0.3, filtration radius r min =1.5, volume constraint f =0.5, convergence condition and other optimization parameters;
5) The initialization unit design variable x is 0.5.
(2) Self-supporting structure topology optimization
Based on a discretized design domain, a self-supporting unit filtering method with a variable critical angle is constructed, self-supporting unit filtering and density filtering are sequentially carried out, the minimized flexibility (maximized rigidity) is taken as a target function, the structural volume fraction is taken as a constraint condition, a topological optimization mathematical model suitable for an additive manufacturing self-supporting structure with the variable critical angle is constructed, and the optimization mathematical model is as follows:
an objective function: minimizing compliance
Constraint conditions are as follows: v (x) P (x))≤f.V 0
Designing variables: x is more than or equal to 0 and less than or equal to 1
Wherein c is the structural flexibility, V is the structural volume, V 0 For the design domain total volume, f is the specified volume fraction, x is the design variable vector, x P Is a physical density vector.
The method comprises the following specific steps:
1) And filtering the self-supporting unit to obtain the self-supporting density of the unit, wherein the expression is as follows:
wherein x is ij Designing variables for the cells;
is a cell free standing density;
for the self-supporting judging factor, the expression is
Where β is an approximate parameter, increasing gradually to 128; t is a threshold value and is set to 0.1；S ij For the unit average density of the supporting unit set, the supporting unit set comprises the target unit x ij Three (two border cells) adjacent cells of the lower layer
Where i and j represent the horizontal and vertical positions of the cell, respectively. A schematic diagram of self-supporting unit judgment at different critical angles is shown in fig. 4;
2) The density filtration obtains the physical density of the unit, and the expression is as follows:
wherein
Is the cell physical density;
is a cell free standing density; n is a radical of hydrogen e Is a unit set in the filtering area; h ei Is a weight coefficient expressed as H ei ＝max(0,r min - Δ (e, i)), wherein r min For the filter radius, Δ (e, i) is the distance between the centers of the cells i and e;
3) Calculating the structure flexibility and the structure volume by finite element analysis, wherein the flexibility expression is as follows:
the volume expression is as follows:
wherein U is an overall displacement matrix; k is an integral rigidity matrix; n is the number of the discretization design domain units; u. u e Is a cell displacement matrix;
is the unit physical density; k is a radical of formula e For a matrix of cell stiffness, the expression is: k is a radical of e ＝E e (x P ).k 0 Wherein k is 0 As an initial cell stiffness matrix, E e (x P ) Is the modulus of elasticity of the unit;
four-node rectangular unit initial unit stiffness matrix k with aspect ratio of 1 0 The expression is:
wherein E is the elastic modulus, mu is the Poisson's ratio,
the relationship between SIMP model cell density and elastic modulus can be expressed as:
wherein E is 1,E min Is taken to be 1 × 10 -9 And P is taken to be 3,x P Is the unit physical density;
obtaining an overall stiffness matrix K through the assembled unit stiffness matrix, establishing a stiffness equation to solve an overall displacement matrix U, wherein the expression is as follows: KU = F, wherein K is an overall stiffness matrix, U is an overall displacement matrix, and F is a load matrix;
4) And sensitivity analysis, wherein the sensitivity expression of the objective function to the design variable is as follows:
the sensitivity expression of the constraint on the design variables is:
where c is the objective function, V is the volume constraint, x is the cell design variable, x F Is a unit self-supporting density, x P Is the cell physical density;
5) Solving an updating unit design variable x by adopting a moving asymptote method (MMA);
6) And calculating the maximum difference value of the updated design variable and the design variable of the previous step, judging whether the maximum difference value is smaller than the convergence value, setting the maximum difference value to be 0.01, if so, terminating the loop iteration, and otherwise, repeating the steps.
(3) Post-treatment
A four-node rectangular cell size is set to a =1,
and extracting a density isosurface based on the optimal layout of the unit density in the design domain to obtain an optimal self-supporting structure design scheme.
Claims (5)
1. A topological optimization method for an additive manufacturing self-supporting structure with variable critical angle is characterized by comprising the following steps:
1) Pretreatment: defining a structural geometric model, dispersing a design domain, adding boundary conditions, initializing design variables and setting optimization parameters; the method comprises the following specific steps:
101 Defining a geometric model of the structure;
102 Setting a critical angle alpha, defining the length and width of the four-node rectangular unit as a and b respectively, and satisfying b = a × tan alpha;
103 A nelx × nely four-node rectangular unit discrete design domain is adopted, if both nelx and nely are integers, the design domain is discrete, otherwise, the nelx and nely are rounded, the length and the width of the unit are adjusted to be a 'and b', the adjusted unit discrete design domain is adopted, and the adjusted critical angle is obtained
104 Adding boundary conditions, applying load F, setting penalty factor P, elastic modulus E, poisson ratio mu, and filtering radius r min Volume constraint f and convergence condition optimization parameters;
105 Initializing a unit design variable x;
2) Self-supporting structure topological optimization: constructing a topological optimization mathematical model of the additive manufacturing self-supporting structure with variable critical angle and solving to obtain the optimal layout of the unit density of the design domain;
3) And (3) post-treatment: and setting the unit size, and extracting a density isosurface to obtain an optimal self-supporting structure design scheme.
2. The topological optimization method for the additive manufacturing self-supporting structure adaptive to the variable critical angle according to claim 1, wherein in step 2), a filtering method for the self-supporting unit with the variable critical angle is constructed based on the discretized design domain, and the filtering method for the self-supporting unit with the variable critical angle and the filtering method for the density are sequentially performed, so that a topological optimization mathematical model for the additive manufacturing self-supporting structure adaptive to the variable critical angle is constructed by taking the minimized flexibility, i.e., the maximized stiffness, as an objective function and taking the structural volume fraction as a constraint condition.
3. The topological optimization method for the additive manufacturing self-supporting structure adaptive to the variable critical angle according to claim 2, wherein the optimization mathematical model is as follows:
an objective function: minimizing compliance
Constraint conditions are as follows: v (x) P (x))≤f.V 0 ；
Designing variables: x is more than or equal to 0 and less than or equal to 1;
wherein c is the structural flexibility, V is the structural volume, V 0 For the design domain total volume, f is the specified volume fraction, x is the design variable vector, x P Is a physical density vector.
4. The topological optimization method for the additive manufacturing self-supporting structure adaptive to the variable critical angle according to claim 3, wherein the specific steps of the step 2) are as follows:
201 Self-supporting unit filtration to obtain unit self-supporting density, the expression is:
wherein x is ij Designing variables for the cells;
is a cell free standing density;
for the self-supporting judging factor, the expression is
Wherein β is an approximation parameter; t is a threshold value; s. the ij For the average density of the support unit set, the support unit set contains target units x ij Three adjacent cells of the lower layer
The number of the boundary units is two, i and j respectively represent the positions of the units in the horizontal and vertical directions;
202 Density filtering to obtain the unit physical density, the expression is:
wherein
Is the unit physical density;
is a cell free standing density; n is a radical of e Is a unit set in the filtering area;H ei is a weight coefficient expressed as H ei ＝max(0,r min - Δ (e, i)), wherein r min For the filter radius, Δ (e, i) is the distance between the centers of the cells i and e;
203 Finite element analysis calculates the structure flexibility and the structure volume, and the flexibility expression is as follows:
the volume expression is as follows:
wherein U is an overall displacement matrix; k is an integral rigidity matrix; n is the number of the discretization design domain units; u. u e Is a cell displacement matrix; k is a radical of e In the form of a matrix of cell stiffness,
is the cell physical density;
204 Sensitivity analysis, the sensitivity expression of the objective function to the design variable is:
the sensitivity expression of the constraint on the design variables is:
where c is the objective function, V is the volume constraint, x is the cell design variable, x F Is a unit self-supporting density, x P Is the cell physical density;
205 Solving the updating unit design variable x by adopting a moving asymptote method;
206 Calculating the maximum difference value between the updated design variable and the design variable of the previous step, judging whether the maximum difference value is smaller than the convergence value, if so, terminating the loop iteration, otherwise, repeating the steps.
5. The topological optimization method for the additive manufacturing self-supporting structure adaptive to the variable critical angle according to claim 4, wherein in the step 3), the four-node rectangular unit size is set, and the density isosurface is extracted based on the optimal layout of the unit density in the design domain to obtain the optimal design scheme for the self-supporting structure.
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