CN107577837B - Structure optimization method for describing class truss by adopting partition interpolation polynomial - Google Patents
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Abstract
The invention discloses a structure optimization method for describing a truss-like structure by adopting a partition interpolation polynomial. The position of the control points and the density of the material at the control points are used as design variables. The material distribution density and direction are calculated according to interpolation polynomials inside each sub-field. These sub-fields are uniformly divided into finite elements. Finite element analysis and optimization analysis were performed. The location of the over-optimized control points and the density of the material at the control points achieve an optimized distribution of the truss-like material. On the basis, the truss can be simply obtained by selecting a plurality of discrete parameters, or the uniform isotropic perforated plate can be obtained by selecting a plurality of parameters of the segmented continuous interval. The truss-like structure optimization is realized by optimizing the shape of the boundary line of the subareas and the distribution of the material on the boundary line of the subareas, so that the problem of structural topology optimization design of the non-uniform anisotropic material is solved.
Description
Technical Field
The invention relates to a structure optimization method for describing a truss-like structure by adopting a partition interpolation polynomial, belongs to the field of structure optimization design, and can be applied to a topological optimization design method for a truss and a porous homogeneous isotropic continuum.
Background
In many fields such as aerospace, materials, machinery, civil engineering, ships and water conservancy, various materials and structures are required to be optimally designed. These materials and structures are not only required to meet the functional requirements of use, but it is also desirable to use as little material as possible. How to optimally design various materials and structures is a crucial issue in these fields. It is not easy to solve these problems. Due to the development of material science and manufacturing technology, the range of materials which can be manufactured is continuously expanded, but how to scientifically and reasonably design materials and structures cannot be well solved. This belongs to the topological optimization design problem of material and structure.
At present, various structural topology optimization methods adopt isotropic material models to directly obtain uniform isotropic continuous bodies with holes. The topological optimization result is expressed by units, and the methods have the problems of unstable numerical calculation such as jagged boundaries, gray density, checkerboard phenomenon and the like and need further processing. More importantly, the topological optimization structure should theoretically be a non-uniform anisotropic truss-like continuum. These methods are far from the theoretical optimal solution. The optimization method based on the truss-like material model can directly obtain an optimized structure which is very close to a theoretical solution.
The traditional optimization method based on the truss-like material model takes the material density and the direction of the node positions of finite elements as optimization design variables. For the problem of minimum stress constrained volume, the density and the direction of the material at the node position are optimized by adopting a full stress criterion, and the density and the direction of the material at any position in the unit are obtained through interpolation, so that an optimized material distribution field is formed. For finite element analysis to be sufficiently accurate, a relatively large number of elements and nodes are generally required. Since the design variables are defined at the node positions, too many nodes means too many design variables, which results in a reduction in optimization efficiency. Particularly, when a sensitivity-based mathematical programming method is adopted to research the general structural topology optimization problem, the accuracy of the sensitivity analysis is low relative to the accuracy of the structural analysis, so that the optimization accuracy is not high. In the optimization process, the direction of the material at each node position is often changed violently, so that a reasonable continuous smooth curve is difficult to form, and the error is large. Further, since the final optimization objective is the material distribution field, the direction of the material at the node position is taken as the optimization design variable, and the re-synthesis of the distribution field function is an indirect method, the workload and the error are increased. It is more reasonable to optimize the distribution function directly.
Disclosure of Invention
The invention provides a structure optimization method for describing a truss-like structure by adopting a partition interpolation polynomial, which solves the optimization design of a non-uniform anisotropic truss-like continuum and finds a topologically optimized rod system structure and a homogeneous isotropic perforated continuum. The technical scheme adopted by the invention for solving the technical problems is as follows:
the method for describing the structure optimization of the class truss by adopting the partition interpolation polynomial comprises the following steps:
the first step is to establish an approximate skew coordinate system O αβ with the load action point O as the origin, the coordinate axis of the curve is approximately represented by a plurality of straight line segments, and the included angle between the 1 st segment of the O α axis and the x axis is α10Length of l11The axis O α starts from segment 2 and makes an angle α with the direction of the previous segmenti0(i-2, 3, …, n) and each segment has a length l1i(i-1, 2, …, n-1) forming a control point Pi0(i is 1,2, …, n) n is the number of O α axis divisions, when the coordinate axis intersects the boundary, e.g., Pn0The length of the line segment is not used as a design variable, the end point of the line segment is selected on the boundary, and the 1 st line segment of the O α axis is rotated by 90 degrees and β degrees in the counter clockwise direction10Is the direction of the 1 st segment of the O β axis and has a length of l21The axis O β starts from segment 2 and makes an angle β with the direction of the previous segment0i(i-2, 3, …, m) and the length of each segment is l2i(i ═ 1,2,3, …, m), forming a control point P0i(i is 1,2,3, …, m), m is the number of division of O β axes, the number of division of n and m of two coordinate axes is preset, the two parameters determine the calculation precision, and when all angles are initially taken as zero, O αβ is a rectangular coordinate system;
the second step is that: establishing a coordinate grid, and dividing a design domain into a plurality of quadrangles; the specific process is as follows: with P10And P01As a starting point, along P10The average direction of the front and the back line segments is further rotated β counterclockwise11Making a straight line; along P01The average direction of the front and the back line segments is further rotated α counterclockwise11Making a straight line, the two straight lines intersecting at P11Form a quadrangle OP10P11P01(ii) a Then with P11And P02Forming the next quadrangle P as a starting point01P11P12P02(ii) a With P02And P11Forming the next quadrangle P as a starting point10P20P21P11(ii) a By analogy, dividing the design domain into intersected grid regions to form a plurality of quadrangles; when all angles are initialWhen the number is zero, the grid is a straight line orthogonal regular grid, the quadrangles are all rectangles,
thirdly, dividing the finite elements in the design domain of the quadrangle to form the basic parameters of the grid αij(i＝1,2,…,n；j＝0,1,2,…,m),βij(i＝0,1,2,…,n-1；j＝0,1,2,…,m),lij(i-1, 2; j-1, 2, …, m-1) and the material density t at the location of the control pointsij(i ═ 0,1,2, …, n; j ═ 0,1,2, …, m) as design variables; the tangential direction of the grid lines is the direction of the material, and the average direction of the front and rear two sections of lines of the control point is taken during actual calculation; with such a grid, a distribution field of the material is actually obtained; the material density of the node position is obtained by interpolation of the shape function of the finite element according to the material density and the direction of the control point position,
in the formula (x)i,yi) Coordinates representing four corner points of the quadrangle, (ξ) is a parameter;
the curve of the material distribution field function formed by the formula (1) represents two material directions, that is, when ξ ═ c, wherein c is constant, η varies as parameter between [ -1,1], the curve represented by the formula (1) represents one material direction of the curve passing through the point, and similarly, when η ═ c, wherein c is constant, ξ varies as parameter between [ -1,1], the curve represented by the formula (1) represents the other material direction, and the material density also adopts an interpolation mode similar to the formula (1)
In the formula tbiRepresenting the material density at the positions of four control points of the quadrangle, thereby determining the material density at the positions of the nodes of the finite element;
the fourth step: finite element analysis, solving node displacement, stress and strain; calculating the sensitivity of the objective function on the design variable by a difference method;
the fifth step: calculating a KKT condition, and stopping calculation if the KKT condition is met; otherwise, entering the next step;
the sixth step is that the objective function and the constraint function are expanded linearly, the design variable is optimized by adopting a quasi-quadratic programming method, and the design variable is controlled αij,βijWithin a smaller range, and returns to the second step.
Compared with the background technology, the technical scheme has the following advantages:
the invention separates the design variables from the nodes of the finite element and divides the design domain into a plurality of quadrilateral subdomains. Each subfield contains a number of finite elements. The material distribution function field within the partition is approximated with a larger grid partition interpolation polynomial. The two opposite side directions of the quadrangle are taken as the distribution directions of the two groups of materials, and the density and the direction of the material in the quadrangle are calculated by means of interpolation polynomial.
The truss-like structure optimization is realized by optimizing the shape of the boundary line of the subareas and the distribution of the material on the boundary line of the subareas, so that the problem of structural topology optimization design of the non-uniform anisotropic material is solved.
Drawings
The invention is further illustrated by the following figures and examples.
FIG. 1 class truss optimized variables and structural topology
FIG. 2 is a flow chart of the optimization method of the present invention.
FIG. 3 shows the initial design domain and loading of a rectangular panel structure according to one embodiment.
Figure 4 is the optimal material distribution for the embodiment obtained with the method of the invention.
Detailed Description
Referring to fig. 1 to 4, a partition interpolation polynomial is used to describe a truss-like structure optimization method for performing a topological optimization design of a cantilever rectangular beam
A rectangular design field of 1.6 meters long L, 1 meter high H, and 0.01 meters thick is shown in fig. 2. The lower left corner position is acted upon by a concentrated force of 100kN vertically downwards and the right side is fixed. The final material optimization distribution visualization graph is shown in fig. 3. And dividing the number of parts in two directions into n and m and 3, and taking a load acting point as a coordinate origin O.
The optimization steps are as follows:
1. preliminary selection of control parameters: initial length in both directions is l1j＝L/n,l2jH/m and direction angle αij＝0,βij＝0,tij＝0.2。
2. The grid is formed according to the control parameters. The length of the grid line in the 3 rd column on the right is not taken as a design variable, but is subject to intersecting with a fixed boundary.
3. And a design domain dividing unit formed by combining all the sub-domains. Within each quadrilateral subdomain, a shape function is selected as the interpolation function. And interpolating according to the material density and the direction of the control points to obtain the material density and the direction of the node position, and forming a structural rigidity matrix.
4. And carrying out finite element analysis to obtain a node displacement column vector. And calculating the strain of the node position and the strain of the main stress direction according to the strain of the node position and the elastic matrix.
5. The derivatives of the objective function with respect to the design variables are calculated using a differential method. And if the KKT optimization condition is met, finishing the optimization, and if not, entering the next step.
6. And linearly expanding the objective function and the constraint function, and optimizing the design variables by adopting a quasi-quadratic programming method. And returning to the step 2.
The above description is only a preferred embodiment of the present invention, and therefore should not be taken as limiting the scope of the invention, which is defined by the appended claims and their equivalents.
Claims (3)
1. The structural optimization method for describing the class truss by adopting the partition interpolation polynomial is characterized by comprising the following steps of:
the first step is to establish an approximate skew coordinate system O αβ with the load action point O as the origin, the coordinate axis of the curve is approximately represented by a plurality of straight line segments, and the included angle between the 1 st segment of the O α axis and the x axis is α10Length of l11The axis O α starts from segment 2 and makes an angle α with the direction of the previous segmenti0(i＝2,3,…,n)By definition, each segment has a length of l1i(i-1, 2, …, n-1) forming a control point Pi0(i is 1,2, …, n) n is the number of O α axis divisions, when the coordinate axis intersects the boundary, e.g., Pn0The length of the line segment is not used as a design variable, the end point of the line segment is selected on the boundary, and the 1 st line segment of the O α axis is rotated by 90 degrees and β degrees in the counter clockwise direction10Is the direction of the 1 st segment of the O β axis and has a length of l21The axis O β starts from segment 2 and makes an angle β with the direction of the previous segment0i(i-2, 3, …, m) and the length of each segment is l2i(i ═ 1,2,3, …, m), forming a control point P0i(i is 1,2,3, …, m), m is the number of division of O β axes, the number of division of n and m of two coordinate axes is preset, the two parameters determine the calculation precision, and when all angles are initially taken as zero, O αβ is a rectangular coordinate system;
the second step is that: establishing a coordinate grid, and dividing a design domain into a plurality of quadrangles; the specific process is as follows: with P10And P01As a starting point, along P10The average direction of the front and the back line segments is further rotated β counterclockwise11Making a straight line; along P01The average direction of the front and the back line segments is further rotated α counterclockwise11Making a straight line, the two straight lines intersecting at P11Form a quadrangle OP10P11P01(ii) a Then with P11And P02Forming the next quadrangle P as a starting point01P11P12P02(ii) a With P02And P11Forming the next quadrangle P as a starting point10P20P21P11(ii) a By analogy, dividing the design domain into intersected grid regions to form a plurality of quadrangles; when all angles are initially taken as zero, the grid is a straight-line orthogonal regular grid, the quadrilaterals are all rectangles,
thirdly, dividing the finite elements in the design domain of the quadrangle to form the basic parameters of the grid αij(i＝1,2,…,n；j＝0,1,2,…,m),βij(i＝0,1,2,…,n-1；j＝0,1,2,…,m),lij(i-1, 2; j-1, 2, …, m-1) and the material density t at the location of the control pointsij(i＝0,1,2,…,n；j＝0,1,2,…,m) As a design variable; the tangential direction of the grid line is the direction of the material, and the average direction of the front section line and the rear section line of the control point is taken during actual calculation; with such a grid, a distribution field of the material is actually obtained; the material density of the node position is obtained by interpolation of the shape function of the finite element according to the material density and the direction of the control point position,
in the formula (x)i,yi) Coordinates representing four corner points of the quadrangle, (ξ) is a parameter;
the curve of the material distribution field function formed by the formula (1) represents two material directions, that is, when ξ ═ c is constant, η is changed between [ -1,1] as a parameter, the curve represented by the formula (1) represents one material direction of the curve passing through points, and similarly, when η ═ c is constant, ξ is changed between [ -1,1] as a parameter, the curve represented by the formula (1) represents the other material direction, and the material density also adopts an interpolation mode similar to the formula (1)
In the formula tbiRepresenting the material density at the positions of four control points of the quadrangle, thereby determining the material density at the positions of the nodes of the finite element;
the fourth step: finite element analysis, solving node displacement, stress and strain; calculating the sensitivity of the objective function on the design variable by a difference method;
the fifth step: calculating a KKT condition, and stopping calculation if the KKT condition is met; otherwise, entering the next step;
the sixth step is that the objective function and the constraint function are expanded linearly, the design variable is optimized by adopting a quasi-quadratic programming method, and the design variable is controlled αij,βijWithin a smaller range, and returns to the second step.
2. The method of claim 1The method for describing the structure optimization of the truss-like structure by using the partition interpolation polynomial is characterized in that in the sixth step, for example, the value of | αij|＜π/2。
3. The method for optimizing the structure of the class truss by using the partition interpolation polynomial as claimed in claim 2, wherein in the sixth step, | αij|≤π/8,|βij|≤π/8。
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| CN105386631A (en) * | 2015-10-10 | 2016-03-09 | 重庆建工住宅建设有限公司 | K-shaped web member lattice type power transmission tower and stability analyzing method thereof |
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