CN115310332A - Porous model compact topology optimization method based on Voronoi division - Google Patents
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Abstract
The invention discloses a porous model compact topology optimization method based on Voronoi division, which directly takes explicit topology and geometric parameters as design freedom and constructs a compact design space for porous model design. Unlike traditional voxel-based model representations, the porous model is represented by a continuous distance field function, enabling refined extraction of model features; and the representation method keeps uniform in the optimization process, and unstable model conversion is avoided. The provided compact topological optimization method can simultaneously optimize the global topological structure and the local size distribution of the porous model according to effective gradient calculation, and can generate high-quality porous models for various complex free surface models. In addition, the invention provides a numerical coarsening method, a general polygon/polyhedron function is constructed, and compared with a classical numerical homogenization method, the method improves the simulation precision by one order of magnitude with similar calculation cost.
Description
Technical Field
The invention relates to the technical field of structure optimization, in particular to a porous model compact topology optimization method based on Voronoi division.
Background
Porous models are widely used in industry with relatively low density and versatile characteristics, and have demonstrated unique and superior industrial value in important equipment or instruments such as aerospace components, medical implant devices, energy absorbing protection devices, and the like. However, due to its extremely complex structure, automated design of porous models with specified target properties presents major challenges. The dual-scale topological optimization method is a porous model design method which is widely researched at present. Under this framework, the design domain is typically discretized into a set of very fine voxel cells to capture the complex structural distribution of the porous model. These discrete elements are used to represent the basic geometry of the model, as well as the design parameters and finite element meshes for performance optimization. Under this representation, the porous model design problem is expressed as an optimal voxel cell distribution under certain physical or geometric constraints to meet specific design goals. This approach involves iterative structural simulation and re-modification until convergence.
For complex porous models, discrete representation based on voxel cells brings a large amount of design freedom, corresponding to dense design space. The inherent challenge is that explicit geometric information of the model is lost during the optimization process, often resulting in an ineffective porous model that cannot be practically manufactured, possibly including broken, elongated or very small pore distributions. Therefore, this approach requires the application of cumbersome geometric constraints to generate efficient models or to meet specific shape control objectives. Also in such a discrete representation, the shape of the structure depends on the resolution of the voxel grid, and it is very challenging to extract fine features in the optimization process. Furthermore, the large number of voxel units as the underlying finite element mesh poses a serious challenge to the computational efficiency of the simulation. The use of classical numerical homogenization methods can improve efficiency but with up to 20 times the loss of accuracy.
Another commonly used method is to embed some type of porous element in accordance with a specified material or physical field. This strategy allows easy control of the model shape, but greatly limits the cell types used, and thus the physical properties that can be achieved by the porous model. The embedded cells may be expanded by inverse homogenization methods or parametric reconstruction processes, and although these methods can produce a large number of porous cells, candidate cells can become very limited when it is desired that neighboring cells remain geometrically connected. More importantly, such methods are generally only applicable to conventional design domains, such as cuboids.
Disclosure of Invention
Aiming at the problems existing in the porous structure design, the invention provides a porous model compact topology optimization method based on Voronoi division. The effectiveness of the porous model design depends largely on the design space in which the porous model is described. An ideal design space needs to satisfy three attributes: explicit geometric control, wide spatial range, and efficient performance simulation. Both the topology and the geometry of the porous model can be adjusted by a well-defined set of geometric parameters, independent of the particular mesh resolution. The parameterization space can cover a wide range of design candidates to meet specific attribute/function requirements. Therefore, based on Voronoi division, explicit topology and geometric control parameters are defined, a compact design space is established to describe the porous model, and a compact topology optimization method for designing the porous model is provided.
The technical scheme adopted by the invention is as follows:
a porous model compact topology optimization method based on Voronoi division comprises the following steps:
1) In a given design domain, voronoi division is carried out according to input seed points, a porous model based on the Voronoi division is constructed, the seed points divided by Voronoi are used as topological control parameters, each edge of the topological control parameter is regarded as a rod, and the rod diameter is used as a geometric control parameter, namely the Voronoi porous model. Further establishing implicit expression for each rod, and performing implicit fitting on all connected rods to obtain an overall smooth continuous natural self-connected porous model;
2) Constructing a general polygon/polyhedron material perception shape function, establishing a mapping relation from unit nodes of a generalized coarse grid, namely curve bridging nodes, to nodes inside the model, and realizing high-precision simulation of the porous model on the coarse grid;
3) Based on a gradient optimization Voronoi porous model, the seed points and the rod diameters divided by Voronoi are used as design variables, physical properties are used as guidance (structural flexibility is minimized), the geometric shape of the porous model is constrained by a classical centroid Voronoi division method, and the porous model topology optimization in a compact design space is realized.
In the above technical solution, the step 1) is specifically as follows:
in a given design domain, randomly sampling to obtain initial seed point distribution, carrying out Voronoi division according to the initial seed points, and enabling each edge to correspond to each rod in the Voronoi porous model, wherein each rod is uniquely determined by one vector, and under the two-dimensional condition, each rod is uniquely determined by one vector
Wherein (x) j ，y j ) Is the center point of the rod, L j Is the length of the rod, alpha j Is the angle of inclination of the bar (counterclockwise about the horizontal x-axis),
is the radius of the rod; in the three-dimensional case
Wherein (x) j ，y j ，z j ) Is the center point of the rod, α j ，θ j ，γ j The included angle between the local coordinate system of the rod and the global coordinate system is determined according to the right hand rule, the long edge of the rod is taken as an x axis, the local coordinate system is established, and the global coordinate system is a real coordinate system;
corresponding each seed point to a rod diameter variable, namely r = { r = { r i N, where N is the number of seed points, and for the j-th edge in the Voronoi division, the width corresponding to the edge is the width of the j-th edge
Determined by the average of the rod diameter variables of its neighboring seed points,
wherein,
is a set of adjacent seed points that are,
k represents the kth adjacent seed point of the jth edge as the number of adjacent seed points.
For any point x in the design domain, the function phi is described by the topology j (x) Each rod is described, resulting in an implicit representation of each rod. The topological description of the overall structure is the union of the topological description functions of all the rods, i.e., Φ (x) = max (Φ) 1 ，...，φ n ) And n is the number of rods. In practical calculations, Φ (x) is approximated by a continuous and smooth function,
where p =2.
From the point of view of numerical implementation, a step function H (phi (x)) is introduced for regularization,
where e is a parameter controlling the regularization size, α =1e -3 Take a very small positive value to avoid global stiffness matrix singularities.
Further, the step 2) constructs a general polygon/polyhedron material perceptual shape function, specifically as follows:
for arbitrary division intoA polygonal or polyhedral finite element mesh D comprising M coarse mesh finite elements D α For each coarse grid cell D α And also carrying out finite element division to obtain m α A fine grid finite element D e . And selecting boundary nodes on part of the fine grids as bridging nodes, inserting two Bezier control points into every two bridging nodes in the polygon unit, inserting polygonal generalized Bezier control points into the polyhedron unit according to the generalized centroid coordinate, wherein the bridging nodes and the Bezier control points are collectively called curve bridging nodes, and the displacement of the curve bridging nodes is used as the degree of freedom of simulation calculation of the coarse grids.
Definition of N α (x) For the shape function to be solved, from Bessel interpolation matrix psi and boundary-inner transformation matrix
The components of the composition are as follows,
wherein N is h (x) Is a classical linear interpolation function. Boundary-interior transformation matrix
The mapping relation between the boundary nodes and the internal nodes can be established by solving the coarse grid unit D α The local finite element problem of (2).
The Bessel interpolation matrix Ψ establishes a mapping relationship between the curve bridging nodes and the boundary nodes. For a polygon element, Ψ is based on the Bessel bases corresponding to all boundary nodes on each edge of the polygon
So as to obtain the compound with the characteristics of,
wherein x b Set of all Bessel control points on edge bBessel bases for all boundary nodes
Arranged in rows.
For a polyhedral unit, each face is a polygon with uncertain number of edges, and omega is defined by representing the polygon by a generalized Bessel surface t (x) Is a generalized centroid basis function of the irregular polygon P, n is the number of all generalized Bessel control points in the polygon P, and for the t control point, the basis function
Is that
Is expanded by the polynomial of (a),
wherein
In order to be a multiple of the index,
degree d =2 in the method for polynomial coefficients.
Similarly, the Bessel interpolation matrix Ψ of a polyhedron unit is composed of Bessel bases corresponding to all boundary nodes on each face of the polyhedron
To obtain
Wherein x P Is the set of all bessel control points on the plane P.
According to the shape function N α (x) Performing high-precision simulation on the coarse grid;
global stiffness matrix K of
Unit stiffness matrix K α In order to realize the purpose,
wherein the penalty factor q =2,H q (Φ (x)) is used to represent the material density at point x, [ LN ] h ]Is N h (x) Partial derivatives with respect to x, D 0 The elasticity tensor of the material is defaulted for the porous model.
Further, for a more complex design domain Ω, such as a curved surface body, based on the shape function constructed in the step 2), an embedded simulation frame is introduced, simulation is performed on a coarsened background grid, dirichlet boundary conditions of the background grid are applied in a weak form based on a Nitsche method, and the simulation method can be expanded to a design domain with a complex free-form surface.
Further, the step 3) is specifically as follows:
the method innovatively takes the seed point X divided by Voronoi and the rod diameter r defined on the point as design variables. In the optimization process, in addition to the objective structure flexibility C (X, r) being desired to be minimum, from the viewpoint of practical manufacturability, a classical centroid Voronoi division energy function S (X) is introduced to control the geometry of Voronoi division so that the inner hole is as close to a regular polygon as possible, noting that the energy function S (X) is only related to the seed point X,
The final optimization problem is expressed as,
wherein Q is a background grid node displacement vector to be calculated, K (X, r) is a porous model integral rigidity matrix, f represents a node external force load vector, w is a weight for controlling the ratio of two objective functions, V is a target volume fraction, and V is a target volume fraction 0 Is the total volume of the design domain omega, V (X, r) is the volume of the Voronoi porous model, C (X, r) is the structural compliance,
C(X，r)＝f T Q
wherein y is α (X, r) is a coarse grid cell D α The volume of material of (a).
The invention has the beneficial effects that:
1) A compact design space is constructed, based on Voronoi division, a wide design space is covered by a small number of parameters, and the topology and the geometric shape of the porous model can be controlled explicitly.
2) The continuous implicit expression of the porous model is established, extremely fine features can be extracted, unified expression is provided for downstream simulation and optimization, and unstable complex model conversion is avoided.
3) A numerical value coarsening method based on a matrix-form polygon/polyhedron-form function is provided, and compared with classical numerical value homogenization, simulation precision is improved by one order of magnitude with similar calculation cost.
4) Through effective gradient calculation, the global topology and the local size distribution of the porous model can be optimized simultaneously.
Drawings
FIG. 1 is a schematic flow diagram of the process of the present invention; (a) problem definition and background meshing → (b) initial seed point distribution → (c) initial Voronoi porous model → (d) simulation of Voronoi porous model → (e) optimization of Voronoi porous model, returning to (b) until convergence.
FIG. 2 is an example of a cantilever porosity model simulation: a. the simulation result of the method; b. simulation results of the classical numerical homogenization method (grey structures are reference results).
Detailed Description
The invention is further described with reference to the following figures and specific embodiments.
The invention provides a porous model compact topology optimization method based on Voronoi division. The effectiveness of the porous model design depends largely on the design space in which the porous model is described. An ideal design space needs to satisfy three attributes: explicit geometric control, extensive design space, and efficient performance simulation. Both the topology and the geometry of the porous model can be adjusted by a well-defined set of geometric parameters, independent of the particular mesh resolution. The parameterized space can encompass a wide range of design candidates to meet specific property/function requirements. Therefore, based on Voronoi division, the invention defines explicit topology and geometric control parameters, establishes compact design space to describe the porous model, and provides a compact topology optimization method for designing the porous model. The method specifically comprises the following steps:
1) Constructing a porous model based on Voronoi division, taking seed points of the Voronoi division as topology control parameters, taking each edge of the model as a rod, and taking the rod diameter as a geometric control parameter
In a given design domain, randomly sampling to obtain initial seed point distribution, carrying out Voronoi division according to the initial seed points, and enabling each edge to correspond to each rod in the Voronoi porous model, wherein the j-th rod is formed by a vector m j Uniquely determined, in two dimensions
Wherein (x) j ，y j ) Is the center point of the rod, L j Is the length of the rod, alpha j Is the angle of inclination of the bar (counterclockwise about the horizontal x-axis),
is the radius of the rod; in the three-dimensional case
Wherein (x) j ，y j ，z j ) Is the center point of the rod, α j ，θ j ，γ j The included angle between the local coordinate system of the rod and the global coordinate system is determined according to the right hand rule, the long edge of the rod is taken as an x axis, the local coordinate system is established, and the global coordinate system is a real coordinate system;
corresponding each seed point to one rod diameter variable, namely r = { r = i N, where N is the number of seed points, and for the j-th edge in the Voronoi division, the width corresponding to the edge is the width of the j-th edge
Determined by the average of the rod diameter variables of its neighboring seed points,
wherein,
is a set of adjacent seed points that are,
k represents the kth adjacent seed point of the jth edge;
for any point x, the function φ is described by the topology j (x) Describing each rod to obtain eachImplicit representation of the root stem. In the two-dimensional case, phi j (x) The definition is that,
wherein q =6, and
in three dimensions, phi j (x) The definition is that,
wherein q =6, and
the topological description of the overall structure is the union of the topological description functions of all the rods, i.e., Φ (x) = max (Φ) 1 ，...，φ n ) And n is the number of rods. In practical calculations, Φ (x) is approximated by a continuous and smooth function,
where p =2.
From the point of view of numerical implementation, a step function H (phi (x)) is introduced for regularization,
where e is a parameter controlling the regularization size, α =1e -3 Take a very small positive value to avoid global stiffness matrix singularities.
2) Constructing a general polygon/polyhedron material perception shape function, and establishing a mapping relation from generalized coarse grid unit nodes, namely curve bridging nodes, to nodes in the model
For finite element mesh D arbitrarily divided into polygon or polyhedron, M coarse mesh finite elements D are contained α For each coarse grid cell D α And also carrying out finite element division to obtain m α A fine grid finite element D e . The method comprises the steps that boundary nodes on part of fine grids are selected as bridging nodes, two Bezier control points are inserted into every two bridging nodes in a polygonal unit, polygonal generalized Bezier control points are inserted into a polyhedral unit according to generalized centroid coordinates, the bridging nodes and the Bezier control points are collectively called curve bridging nodes, and displacement of the curve bridging nodes is used as the degree of freedom of simulation calculation of the coarse grids.
Definition of N α (x) For the shape function to be solved, from Bessel interpolation matrix psi and boundary-interior transformation matrix
The components of the composition are as follows,
wherein N is h (x) Is a classical linear interpolation function. Boundary-interior transformation matrix
The mapping relation between the boundary nodes and the internal nodes can be established by solving the coarse grid unit D α The local finite element problem of (2).
The Bessel interpolation matrix Ψ establishes a mapping relationship between the curve bridging nodes and the boundary nodes. For a polygon cell, Ψ bases the Bessel bases corresponding to all boundary nodes on each edge of the polygon
So as to obtain the compound with the characteristics of,
wherein x b Bezier bases of all boundary nodes for the set of all Bezier control points on edge b
Arranged in rows.
For a polyhedral unit, each face is a polygon with uncertain number of edges, and omega is defined by representing the polygon by a generalized Bessel surface t (x) Is a generalized centroid basis function of the irregular polygon P, n is the number of all generalized Bessel control points in the polygon P, and for the t control point, the basis function
Is that
Is expanded by the polynomial of (a),
wherein
In order to be a multiple of the index,
degree d =2 in the method for polynomial coefficients.
Similarly, the Bezier interpolation matrix Ψ for a polyhedron unit consists of Bezier bases corresponding to all boundary nodes on each face of the polyhedron
To obtain
Wherein x P Is the set of all bessel control points on the plane P.
According to the shape function N α (x) Performing high-precision simulation on the coarse grid;
global stiffness matrix K of
Unit stiffness matrix K α In order to realize the purpose,
wherein the penalty factor q =2,H q (Φ (x)) is used to represent the material density at point x, [ LN ] h ]Is N h (x) Partial derivatives of x, D 0 The elasticity tensor of the material is defaulted for the porous model.
Further, for a more complex design domain Ω, such as a curved surface body, based on the shape function constructed in step 2), an embedded simulation framework is introduced, simulation is performed on a coarsened background grid, dirichlet boundary conditions of the background grid are applied in a weak form based on a Nitsche method, and the simulation method can be expanded to the design domain with the complex free-form curved surface.
3) Based on the gradient optimization Voronoi porous model, the seed points and the rod diameters divided by the Voronoi are used as design variables, the structural flexibility is minimized, and the geometric shape of the porous model is constrained
Taking the seed point X of the Voronoi partition and the rod diameter r defined at the point as design variables, in the optimization process, in addition to the desired minimum target structure compliance C (X, r), for practical manufacturability reasons, a classical centroid Voronoi partition energy function S (X) is introduced to control the geometry of the Voronoi partition so that the inner hole is as close to a regular polygon as possible, noting that the energy function S (X) is only related to the seed point X,
The final optimization problem is expressed as,
wherein Q is a background grid node displacement vector to be calculated, K (X, r) is a porous model integral rigidity matrix, f represents a node external force load vector, w is a weight for controlling the ratio of two objective functions, V is a target volume fraction, and V is a target volume fraction 0 Is the total volume of the design domain omega, V (X, r) is the volume of the Voronoi porous model, C (X, r) is the structural compliance,
C(X，r)＝f T Q
wherein V α (X, r) is a coarse grid cell D α The volume of material of (a).
The sensitivity of the objective function C (X, r) is derived from the chain rule as follows:
wherein a may be X i Or r i ，
And the number of the first and second groups,
here partial derivatives
The calculation of (c) is obtained by difference.
The sensitivity of the other functions S (X), V (X, r) with respect to design variables is,
wherein
For coarse grid cells D α The unit volume of (a).
The following is a concrete solving case carried out by adopting the method of the invention:
fig. 1 (a) is a femur model with a complex shell obtained by real three-dimensional scanning, the left end is forced upwards, the right end is forced downwards, and the lower end is fixed. (b) The initial randomly sampled seed point distribution, and (c) the initial Voronoi multi-hole model corresponding to the seed point. According to the constructed polyhedron material perceptual shape function and the embedded simulation framework, the simulation deformation result is shown in (d). And (e) optimizing the Voronoi porous model under a certain volume constraint by taking the seed points and the rod diameters distributed on the seed points as design variables and taking the minimized structural flexibility as a target (maximized rigidity), obtaining new seed points and rod diameters after each iteration step is finished, and returning to the step (b) until convergence to obtain the finally optimized porous model. The result can satisfy the design requirement that the femur effectively supports the human body under light weight. According to the method, the porous model is described based on Voronoi division, the physical performance and the porous geometric shape are optimized, natural self-connection of the structure can be guaranteed, and the porous model with optimized physical performance and controllable geometry is obtained finally.
Fig. 2 is a cantilever beam cell model, with the left end fixed to the wall and the right lower end subjected to a downward pulling force. (a) For the simulation result of the method, the error is 5e -3 And (b) is a simulation result of a classical value homogenization method, the error is 0.6, and a gray structure is a reference result.
The invention directly takes explicit topology and geometric parameters as design freedom, constructs compact design space and is used for porous model design. Unlike traditional voxel-based model representations, the porous model is represented by a continuous distance field function, enabling refined extraction of model features; and the representation method keeps uniform in the optimization process, and unstable model conversion is avoided. The provided compact topology optimization method can simultaneously optimize the global topology structure and the local size distribution of the porous model according to effective gradient calculation, and can generate high-quality porous models for various complex free surface models. In addition, the invention provides a numerical coarsening method, a general polygon/polyhedron function is constructed, and compared with a classical numerical homogenization method, the method improves the simulation precision by one order of magnitude with similar calculation cost.
Claims (4)
1. A porous model compact topology optimization method based on Voronoi division is characterized by comprising the following steps:
1) In a given design domain, carrying out Voronoi division according to input seed points, constructing a porous model based on the Voronoi division, taking the seed points divided by the Voronoi as topological control parameters, regarding each edge as a rod, taking the rod diameter as a geometric control parameter, establishing implicit expression for each rod, and carrying out implicit fitting on all connected rods to obtain an integral smooth continuous natural self-connected porous model;
2) Constructing a general polygon/polyhedron material perception shape function, establishing a mapping relation from unit nodes of a generalized coarse grid, namely curve bridging nodes, to nodes inside the model, and realizing high-precision simulation of the porous model on the coarse grid;
3) Based on a gradient optimization Voronoi porous model, the seed points and rod diameters divided by the Voronoi are used as design variables, physical properties are used as guidance, namely, the structural flexibility is minimized, the geometrical shape of the porous model is constrained by a classical centroid Voronoi division method, and the topological optimization of the porous model in a compact design space is realized.
2. The porous model compact topology optimization method based on Voronoi division according to claim 1, wherein the step 1) is as follows:
in a given design domain, randomly sampling to obtain initial seed point distribution, carrying out Voronoi division according to the initial seed points, and enabling each edge to correspond to each rod in the Voronoi porous model, wherein the j-th rod is formed by a vector m j Uniquely determined, in two dimensions
Wherein (x) j ,y j ) Is the center point of the rod, L j Is the length of the rod, alpha j Is the angle of inclination of the bar (counterclockwise about the horizontal x-axis),
is the radius of the rod; in the three-dimensional case
Wherein (x) j ,y j ,z j ) Is the center point of the rod, α j ,θ j ,γ j The included angle between the local coordinate system and the global coordinate system of the rod is determined according to the right hand rule, the long edge of the rod is taken as an x axis, the local coordinate system is established, and the global coordinate system is a real coordinate system;
corresponding each seed point to a rod diameter variable, namely r = { r = { r i I =1 … N }, where N is the number of seed points, and for the j-th edge in the Voronoi partition, the width corresponding to the edge is the width of the j-th edge
Determined by the average of the rod diameter variables of its neighboring seed points,
wherein,
is a set of adjacent seed points that are,
k represents the kth adjacent seed point of the jth edge;
for any point x, the function φ is described by the topology j (x) Describing each rod to obtain an implicit expression of each rod, wherein the topological description of the overall structure is the union of topological description functions of all the rods, namely phi (x) = max (phi) 1 ,…,φ n ) N is the number of rods; in practical calculations, Φ (x) is approximated by a continuous and smooth function,
wherein p =2;
from the point of view of numerical implementation, a step function H (phi (x)) is introduced for regularization,
where e is a parameter controlling the regularization size, α =1e -3 Take a very small positive value to avoid global stiffness matrix singularities.
3. The porous model compact topology optimization method based on Voronoi division according to claim 1, characterized in that the generalized polygon/polyhedron material perceptual shape function is constructed in the step 2), specifically as follows:
for finite element mesh D arbitrarily divided into polygons or polyhedrons, M coarse mesh finite elements D are contained α For each coarse grid cell D α And also carrying out finite element division to obtain m α A fine grid finite element D e (ii) a Selecting boundary nodes on part of the fine grids as bridging nodes, inserting two Bezier control points into every two bridging nodes in a polygon unit, and inserting multi-edge generalized Bezier control points into a polyhedron unit according to generalized centroid coordinates; the bridge nodes and the Bessel control points are called curve bridge nodes together, and the displacement of the curve bridge nodes is used as the degree of freedom of simulation calculation of the coarse grid;
definition of N α (x) For the shape function to be solved, from Bessel interpolation matrix psi and boundary-inner transformation matrix
The components of the components are as follows,
wherein N is h (x) Is a classical linear interpolation function, boundary-internal transformation matrix
Establishing a mapping relation between the boundary nodes and the internal nodes, and solving a coarse grid unit D α The local finite element problem of (2) is obtained;
the Bezier interpolation matrix psi establishes a mapping relation between curve bridging nodes and boundary nodes, and for a polygon unit psi establishes a mapping relation between curve bridging nodes and boundary nodes according to Bezier bases corresponding to all boundary nodes on each edge of the polygon
So as to obtain the compound with the characteristics of,
wherein x b Bessel bases of all boundary nodes for the set of all Bessel control points on edge b
Arranging according to rows;
for a polyhedral unit, each face is a polygon with uncertain number of edges, and omega is defined by representing the polygon by a generalized Bessel surface t (x) Is a generalized centroid basis function of the irregular polygon P, n is the number of all generalized Bessel control points in the polygon P, and for the t control point, the basis function
Is that
Is expanded by the polynomial of (a),
wherein
In order to be a multiple of the index,
is a polynomial coefficient, degree d =2;
similarly, the Bessel interpolation matrix Ψ of a polyhedron unit is composed of Bessel bases corresponding to all boundary nodes on each face of the polyhedron
To obtain
Wherein x P Is the set of all Bessel control points on the plane P;
according to the shape function N α (x) Performing high-precision simulation on the coarse grid;
global stiffness matrix K of
Unit stiffness matrix K α In order to realize the purpose of the method,
wherein the penalty factor q =2,H q (Φ (x)) is used to represent the material density at point x, [ LN ] h ]Is N h (x) About partial derivatives, D 0 Defaulting the elasticity tensor of the material for the porous model;
for a more complex design domain omega such as a curved surface body, an embedded simulation framework can be introduced for simulation.
4. The porous model compact topology optimization method based on Voronoi division according to claim 1, wherein the step 3) is as follows:
taking the seed points of the Voronoi division and the rod diameters defined at the points as design variables, in the optimization process, in addition to the expectation that the target structure compliance C (X, r) is minimum, the classical centroid Voronoi division energy function S (X) is introduced to control the geometry of the Voronoi division so that the inner hole is as close to a regular polygon as possible, the energy function S (X) is only related to the seed points X,
the final optimization problem is expressed as,
wherein Q is a background grid node displacement vector to be calculated, K (X, r) is a porous model integral rigidity matrix, f represents a node external force load vector, w is a weight for controlling the ratio of two objective functions, V is a target volume fraction, and V is a target volume fraction 0 Is the total volume of the design domain omega, V (X, r) is the volume of the Voronoi porous model, C (X, r) is the structural compliance,
C(X,r)＝f T Q
wherein V α (X, r) is a coarse grid cell D α The volume of material of (a).
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