CN107577837A - Structural optimization method for truss-like description using partition interpolation polynomials - Google Patents

Abstract

The invention discloses a structural optimization method for describing a quasi-truss by using partition interpolation polynomials. Firstly, the design domain is divided into several subdomains, and polynomial interpolation functions are respectively constructed in each subdomain according to domain boundary control points. The location 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 the interpolation polynomial within each sub-domain. Generally, these sub-domains are uniformly divided into finite elements. Perform finite element analysis and optimization analysis. The optimal distribution of truss-like materials is achieved by optimizing the position of control points and the density of materials at control points. On this basis, the truss can be simplified by selecting several discretized parameters, or the homogeneous isotropic perforated plate can be obtained by selecting several segmented continuous interval parameters. The truss-like structure is optimized by optimizing the shape of the partition boundary line and the distribution of materials on the partition boundary line, which is used to solve the structural topology optimization design problem of non-uniform anisotropic materials.

Description

Structural optimization method for truss-like description using partition interpolation polynomials

technical field

The invention relates to a structural optimization method for truss-like descriptions using partition interpolation polynomials, which belongs to the field of structural optimization design and can be applied to topology optimization design methods for trusses and homogeneous isotropic continuums with holes.

Background technique

In many fields such as aerospace, materials, machinery, civil engineering, ships and water conservancy, it is often necessary to optimize the design of various materials and structures. These materials and structures not only need to meet the requirements of use functions, but also hope to use as few materials as possible. How to optimally design various materials and structures is a crucial issue in these fields. And solving these problems is not an easy task. Due to the development of material science and manufacturing technology, the range of materials we can manufacture has been continuously expanded, but how to design materials and structures scientifically and rationally has not been well resolved. This belongs to the topology optimization design problem of materials and structures.

At present, various structural topology optimization methods use isotropic material models to directly obtain uniform isotropic porous continuum. The results of topology optimization are represented by units. This kind of method has numerical calculation instability problems such as sawtooth boundary, gray density and checkerboard phenomenon, which need further processing. More importantly, the topology optimized 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 the optimized structure which is very close to the theoretical solution.

The traditional optimization method based on the truss-like material model takes the material density and direction of the finite element node position as the optimization design variables. For the stress-constrained volume minimum problem, the full stress criterion is used to optimize the density and direction of the material at the node position, and the density and direction of the material at any position in the unit are obtained by interpolation, thereby forming an optimized material distribution field. In order to be accurate enough for finite element analysis, more 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 leads to a decrease in optimization efficiency. Especially when the sensitivity-based mathematical programming method is used to study the general structural topology optimization problem, the precision of the sensitivity analysis is lower than that of the structural analysis, resulting in a low precision of the optimization. During the optimization process, the direction of the material at each node position often changes drastically, and it is difficult to form a reasonable continuous smooth curve, and the error is large. Furthermore, since the final optimization goal is the material distribution field, the direction of the material at the node position is used as the optimization design variable, and then the synthesis of the distribution field function is an indirect method, which increases the workload and error. It would be more reasonable to directly optimize the distribution function.

Contents of the invention

The invention provides a structure optimization method using partition interpolation polynomials to describe quasi-truss, solves the optimization design of non-uniform anisotropic quasi-truss continuum, and searches for topology-optimized bar structure and homogeneous isotropic continuum with holes. The technical solution adopted by the present invention to solve its technical problems is:

A structural optimization method for truss-like descriptions using partition interpolation polynomials, including:

The first step: take the load action point O as the origin, establish an approximate oblique coordinate system Oαβ; the curve coordinate axis is approximately represented by multiple straight line segments; the angle between the first segment of the Oα axis and the x-axis is α ₁₀ , and the length is l ₁₁ ; The Oα axis starts from the second line segment and is defined by the direction angle α _i0 (i=2,3,…,n) with the previous line segment, and the length of each line segment is l _1i (i=1,2,…, n-1), forming the control point P _i0 (i=1,2,...,n), n is the number of divisions of the Oα axis; when the coordinate axis intersects with the boundary, such as P _n0 point, the length of the line segment is not used as The design variable, the end point of the line segment is selected on the boundary; the first line segment of the Oα axis is rotated 90 degrees counterclockwise and β ₁₀ is the direction of the first line segment of the Oβ axis, and the length is l ₂₁ ; the Oβ axis starts from the second line segment, with The direction angle β _0i (i=2,3,…,m) of the previous line segment is defined, and the length of each line segment is l _2i (i=1,2,3,…,m), forming the control point P _0i ( i=1,2,3,...,m), m is the number of divisions of the Oβ axis; the number of divisions n and m of the two coordinate axes are set in advance, and these two parameters determine the calculation accuracy; when all angles are initially When taken as zero, Oαβ is the Cartesian coordinate system;

Step 2: Establish a coordinate grid and divide the design domain into several quadrilaterals; the specific process is: starting from P ₁₀ and P ₀₁ , turn β ₁₁ counterclockwise along the average direction of the two segments before and after P ₁₀ to make a straight line; Turn α ₁₁ counterclockwise along the average direction of the two line segments before and after P ₀₁ to make a straight line. These two straight lines intersect at P ₁₁ to form a quadrilateral OP ₁₀ P ₁₁ P ₀₁ ; then form the next quadrilateral with P ₁₁ and P ₀₂ as starting points P ₀₁ P ₁₁ P ₁₂ P ₀₂ ; Take P ₀₂ and P ₁₁ as the starting point to form the next quadrilateral P ₁₀ P ₂₀ P ₂₁ P ₁₁ ; and so on, divide the design domain into intersecting grid areas to form several quadrilaterals; When all angles are initially taken as zero, the grid is a straight-orthogonal regular grid and the quadrilaterals are all rectangles.

Step 3: Divide the finite elements with the design domain composed of the above quadrilaterals; to form the basic parameters of the grid, α _ij (i=1,2,…,n; j=0,1,2,…,m), β _ij (i=0,1,2,…,n-1; j=0,1,2,…,m), l _ij (i=1,2; j=1,2,…,m-1) And the material density t _ij (i=0,1,2,...,n; j=0,1,2,...,m) at the position of the control point is used as the design variable; the grid line and the interpolation function in it are the direction of the material , in the actual calculation, the average value of the direction of the two lines before and after the control point is taken; with such a grid, the distribution field of the material is actually obtained; the material density at the node position is determined by the shape function of the finite element according to the position of the control point The material density and orientation interpolation of ,

In the formula, ( _xi , y _i ) represent the coordinates of the four corner points of the quadrilateral, and (ξ, η) can be understood as parameters;

Formula (1) constitutes the material distribution field function; That is, when ξ=c constant, when η changes between [-1,1] as parameter, the curve represented by formula (1) has represented the material of the point that curve passes through direction; similarly, when η=c constant, ξ as a parameter changes between [-1,1], the curve represented by formula (1) represents another material direction. The material density also adopts the interpolation method similar to formula (1)

In the formula, t _bi represents the material density at the four control points of the quadrilateral, from which the material density at the node position of the finite element can be determined;

The fourth step: finite element analysis, solve the node displacement, stress and strain; calculate the sensitivity of the objective function with respect to the design variables by the difference method;

Step 5: Calculate the KKT condition, and stop the calculation if it is met; otherwise, go to the next step;

Step 6: Expand the objective function and constraint function linearly, and use the quasi-quadratic programming method to optimize the design variables; control the variation range of the design variables α _ij and β _ij within a small range, and return to the second step.

Compared with the background technology, this technical solution has the following advantages:

The invention separates the design variables from the nodes of the finite element, and divides the design domain into several quadrilateral sub-domains. Each subdomain contains several finite elements. The material distribution function field within the partition is approximated by the interpolation polynomial of the larger grid partition. The directions of the opposite sides of the two groups of quadrilaterals are used as the distribution directions of the two groups of materials, and the density and direction of the materials inside the quadrilaterals are calculated by means of interpolation polynomials.

The truss-like structure is optimized by optimizing the shape of the partition boundary line and the distribution of materials on the partition boundary line, which is used to solve the structural topology optimization design problem of non-uniform anisotropic materials.

Description of drawings

The present invention will be further described below in conjunction with drawings and embodiments.

Fig.1 Class truss optimization variables and structure topology

Fig. 2 is a flowchart of the optimization method of the present invention.

Figure 3 is an example of the initial design domain and loads for a rectangular slab structure.

Fig. 4 is the best material distribution of the examples obtained by the method of the present invention.

detailed description

Please refer to Figure 1 to Figure 4, using the partition interpolation polynomial to describe the structural optimization method of the quasi-truss, and carry out the topology optimization design of the cantilever rectangular beam

A rectangular design domain with length L=1.6m, height H=1m, and thickness 0.01m is shown in Figure 2. A 100kN vertical downward concentrated force acts on the lower left corner, and the right side is fixed. The final material optimization distribution visualization graph is shown in Figure 3. The number of divisions in the two directions is n=m=3, and the load action point is taken as the coordinate origin O.

The optimization steps are as follows:

1. Preliminary selection of control parameters: the initial lengths of the two directions are l _1j =L/n, l _2j =H/m and the direction angles α _ij =0, β _ij =0, t _ij =0.2.

2. Form the grid according to the control parameters. The length of the grid line in the third column on the right is not used as a design variable, but is subject to the intersection with the fixed boundary.

3. The design domain division unit formed by combining all sub-domains. Within each quadrilateral subdomain, a shape function is chosen as the interpolation function. According to the material density and direction interpolation of the control points, the material density and direction of the joint position are obtained, and the structural stiffness matrix is formed.

4. Carry out finite element analysis to obtain the node displacement column vector. The principal stress direction of the joint position and the strain in the principal stress direction are calculated from the strain at the joint position and the elastic matrix.

5. The differential method is used to calculate the derivative of the objective function with respect to the design variables. If the KKT optimization conditions are met, the optimization ends, otherwise, it goes to the next step.

6. Expand the objective function and constraint function linearly, and use the quasi-quadratic programming method to optimize the design variables. Return to step 2.

The above is only a preferred embodiment of the present invention, so the scope of implementation of the present invention cannot be limited accordingly, that is, equivalent changes and modifications made according to the patent scope of the present invention and the content of the specification should still be covered by the present invention within range.

Claims (3)

1. A method for structural optimization of class trusses described by partition interpolation polynomials, characterized in that it comprises: The first step: take the load action point O as the origin, establish an approximate oblique coordinate system Oαβ; the curve coordinate axis is approximately represented by multiple straight line segments; the angle between the first segment of the Oα axis and the x-axis is α ₁₀ , and the length is l ₁₁ ; The Oα axis starts from the second line segment and is defined by the direction angle α _i0 (i=2,3,…,n) with the previous line segment, and the length of each line segment is l _1i (i=1,2,…, n-1), forming the control point P _i0 (i=1,2,...,n), n is the number of divisions of the Oα axis; when the coordinate axis intersects with the boundary, such as P _n0 point, the length of the line segment is not used as The design variable, the end point of the line segment is selected on the boundary; the first line segment of the Oα axis is rotated 90 degrees counterclockwise and β ₁₀ is the direction of the first line segment of the Oβ axis, and the length is l ₂₁ ; the Oβ axis starts from the second line segment, with The direction angle β _0i (i=2,3,…,m) of the previous line segment is defined, and the length of each line segment is l _2i (i=1,2,3,…,m), forming the control point P _0i ( i=1,2,3,...,m), m is the number of divisions of the Oβ axis; the number of divisions n and m of the two coordinate axes are set in advance, and these two parameters determine the calculation accuracy; when all angles are initially When taken as zero, Oαβ is the Cartesian coordinate system; Step 2: Establish a coordinate grid and divide the design domain into several quadrilaterals; the specific process is: starting from P ₁₀ and P ₀₁ , turn β ₁₁ counterclockwise along the average direction of the two segments before and after P ₁₀ to make a straight line; Turn α ₁₁ counterclockwise along the average direction of the two line segments before and after P ₀₁ to make a straight line. These two straight lines intersect at P ₁₁ to form a quadrilateral OP ₁₀ P ₁₁ P ₀₁ ; then form the next quadrilateral with P ₁₁ and P ₀₂ as starting points P ₀₁ P ₁₁ P ₁₂ P ₀₂ ; Take P ₀₂ and P ₁₁ as the starting point to form the next quadrilateral P ₁₀ P ₂₀ P ₂₁ P ₁₁ ; and so on, divide the design domain into intersecting grid areas to form several quadrilaterals; When all angles are initially taken as zero, the grid is a straight-orthogonal regular grid and the quadrilaterals are all rectangles. Step 3: Divide the finite elements with the design domain composed of the above quadrilaterals; to form the basic parameters of the grid, α _ij (i=1,2,…,n; j=0,1,2,…,m), β _ij (i=0,1,2,…,n-1; j=0,1,2,…,m), l _ij (i=1,2; j=1,2,…,m-1) And the material density t _ij (i=0,1,2,...,n; j=0,1,2,...,m) at the position of the control point is used as the design variable; the grid line and the interpolation function in it are the direction of the material , in the actual calculation, the average value of the direction of the two lines before and after the control point is taken; with such a grid, the distribution field of the material is actually obtained; the material density at the node position is determined by the shape function of the finite element according to the position of the control point The material density and orientation interpolation of , In the formula, ( _xi , y _i ) represent the coordinates of the four corner points of the quadrilateral, and (ξ, η) can be understood as parameters; Formula (1) constitutes the material distribution field function; That is, when ξ=c constant, when η changes between [-1,1] as parameter, the curve represented by formula (1) has represented the material of the point that curve passes through direction; similarly, when η=c constant, ξ as a parameter changes between [-1,1], the curve represented by formula (1) represents another material direction. The material density also adopts the interpolation method similar to formula (1) In the formula, t _bi represents the material density at the four control points of the quadrilateral, from which the material density at the node position of the finite element can be determined; The fourth step: finite element analysis, solve the node displacement, stress and strain; calculate the sensitivity of the objective function with respect to the design variables by the difference method; Step 5: Calculate the KKT condition, and stop the calculation if it is met; otherwise, go to the next step; Step 6: Expand the objective function and constraint function linearly, and use the quasi-quadratic programming method to optimize the design variables; control the variation range of the design variables α _ij and β _ij within a small range, and return to the second step. 2. The method for structural optimization using partition interpolation polynomials to describe quasi-trusses according to claim 1, characterized in that: in the sixth step, such as |α _ij |<π/2. 3. The method for structural optimization using partition interpolation polynomials to describe quasi-trusses according to claim 2, characterized in that: in the sixth step, |α _ij |≤π/8, |β _ij |≤π/8.