Parameterized level set blade topological optimization method based on conformal geometry theory and isogeometric analysis

Abstract

The invention discloses a parameterization level set blade topological optimization method based on a conformal geometry theory and isogeometric analysis, and the method comprises the following steps: building an initial blade model according to an existing blade profile; performing fluid-solid coupling mechanical analysis on the blade structure model, and calculating a blade surface load; the blade structure model is mapped into a two-dimensional rectangular domain through conformal mapping, and a standard orthogonal grid with the same specification as a mapping grid is established; establishing an optimized mathematical model of the blade structure; isogeometric analysis of the blade structure; optimizing a parameterized level set; and inversely mapping the blade structure to complete the optimization process. According to the method, through the conformal mapping theory and the isogeometric level set method, three-dimensional curved surface dimension reduction processing is achieved, the calculation cost is reduced, and the optimization efficiency is improved. And meanwhile, the high-precision calculation advantage of isogeometric analysis is combined, a level set optimized smooth structure is also provided, the checkerboard problem of methods similar to variable density and the like is avoided, and the purpose of efficient and accurate optimization of the curved surface structure is achieved.

Description

Technical Field

[0001] The invention relates to the technical field of fluid machinery structure optimization design, and in particular to a parameterized level set blade topology optimization method based on conformal geometry theory and isogeometric analysis. Background Art

[0002] The fossil energy crisis is a major problem faced by all countries in the new century, so finding new alternative energy sources is an urgent task. With the rapid growth of demand for new energy power generation technology, the application prospects of turbines and wind turbines have received more and more attention in the field of new energy. As a key component, the optimization of the blade airfoil is very important.

[0003] Topology optimization design is widely used in the design process of blades. Topology optimization design is a design method with optimization as the goal. By adjusting the position and shape of the material, the optimal design of the structure is achieved with the goal of minimizing the weight of the structure and maximizing the stiffness of the structure. Topology optimization design can achieve lightweight blades, improve the rigidity of blades, and improve the fluid dynamics performance of blades. However, in the application process, it is necessary to fully consider the engineering practicality and feasibility, and combine it with other methods to achieve the optimal design solution. With the continuous development of science and technology, the application of topology optimization design in blade design will be further deepened and expanded. The existing blade design and the existing design methods of blades rely too much on experience, the design process is cumbersome, and it has to go through multiple trial calculations. The design is time-consuming and labor-intensive, the design results are not guaranteed, and the calculation efficiency and accuracy are low.

[0004] The information disclosed in this background technology section is only intended to enhance the understanding of the overall background of the invention and should not be regarded as an acknowledgment or any form of suggestion that the information constitutes the prior art already known to a person skilled in the art. Summary of the invention

[0005] The purpose of the present invention is to provide a parameterized level set blade topology optimization method based on conformal geometry theory and isogeometric analysis, so as to overcome the shortcomings of existing blade design such as low computational efficiency and low precision.

[0006] To achieve the above object, the present invention provides a parameterized level set blade topology optimization method based on conformal geometry theory and isogeometric analysis, comprising the following steps:

[0007] Step 1: Establish an initial blade model based on the existing blade shape;

[0008] Step 2: Perform fluid-solid coupling mechanical analysis on the blade structure model to calculate the blade surface load;

[0009] Step 3: The blade structure model is conformally mapped into a two-dimensional rectangular domain, and a standard orthogonal grid with the same specifications as the mapped grid is established;

[0010] Step 4: Establish an optimized mathematical model of the blade structure;

[0011] Step 5: Isogeometric analysis of blade structure;

[0012] Step 6: Parameterized level set optimization;

[0013] Step 7: Inverse map the blade structure to complete the optimization process.

[0014] Preferably, in the above technical solution, step 1 of establishing the initial blade model includes: determining the design variables and constraints according to the design requirements to construct the objective function, and establishing the blade airfoil mathematical model:

[0015]

[0016] Where is the Heaviside function, is the objective function, E is the elasticity tensor, ε is the strain, u is the displacement, v is the virtual displacement belonging to space u, and the inequality V(Ω) represents the volume constraint.

[0017] Preferably, in the above technical solution, step 2 of performing fluid-solid coupling mechanical analysis on the blade structure model includes: dividing the blade structure model into blade grids, setting the boundary definition of the inlet flow and velocity vector, and calculating the three-dimensional pressure of the fluid domain where the blade is located through fluid-solid coupling analysis.

[0018] Preferably, in the above technical solution, step three establishes a standard orthogonal grid with the same specifications as the mapping grid, including: mapping the three-dimensional blade model into a triangulation of a two-dimensional rectangular domain, completing the mapping of the Riemann surface to the Euclidean plane, that is, the local scaling-conformal factor, performing barycentric interpolation on the basis of the triangulation, and establishing a standard orthogonal grid with the same specifications as the domain mapping grid,

[0019] Preferably, in the above technical solution, step three of establishing a standard orthogonal grid having the same specifications as the mapping grid comprises:

[0020] Map the three-dimensional blade structure to the unit open disk D. For the mapping f:M→D, M and f can be calculated by solving the following Laplace equation:

[0021]

[0022] The Riemann metric is given by the arc length parameterization, defined as g = e 2ω(u,v) (dx 0 2 +dy 0 2 ), where (x0 ,y 0 ) is the isothermal coordinate; the smooth mapping f:M→N is a scaling transformation, which is determined by the conformal factor scalar function e 2ω The local scaling operation of the control is performed, and then the mapped two-dimensional grid is established as an orthogonal grid.

[0023] Preferably, in the above technical solution, step 4 of establishing an optimized mathematical model of the blade structure includes: constructing an objective function minimum compliance according to the calculated blade surface load and volume constraints, and establishing an optimized mathematical model of the blade.

[0024] Preferably, in the above technical solution, step 5 of isogeometric analysis of the blade structure includes:

[0025] On the basis of the orthogonal grid, control points and weights are designed, and the B-spline surface is defined by basis functions and control points. The B-spline surface is defined as:

[0026]

[0027] Where N i,p (u), N j,p (v) is the p-order B-spline basis function defined on the knot vectors U and V, controlling the fixed point P i,j .

[0028] Preferably, in the above technical solution, step six parameterized level set optimization includes: introducing the parameterized level set function of isogeometric analysis, combining conformal binding theory, deriving the control equation, the modified Hamilton-Jacobi equation, and then evolving the level set function;

[0029] According to the parameterized level set method

[0030]

[0031] Introducing parameterized level set functions for isogeometric analysis

[0032]

[0033] where φ i (t) is the expansion coefficient associated with the ith grid point, N i (x) is the corresponding basis function;

[0034] According to formulas (4) and (5), the Hamilton-Jacobi equation can be updated as

[0035]

[0036] Combining the conformal geometry theory, the modified isogeometry-conformal geometry-level set function is expressed as follows according to formula (2):

[0037]

[0038] Among them, v is related to the time derivative of the expansion coefficient n It is expressed as:

[0039]

[0040] According to the sensitivity analysis, the normal velocity field is constructed, and the level set function is evolved by solving the Hamilton-Jacobi equation in two-dimensional space.

[0041] Preferably, in the above technical solution, step seven of inverse mapping the blade structure includes: after completing the optimization cycle, a two-dimensional optimized structure is obtained, and then the two-dimensional grid is inversely mapped to the three-dimensional blade surface through barycentric interpolation to complete the optimization process.

[0042] Compared with the prior art, the present invention has the following beneficial effects:

[0043] (1) The present invention is based on a parameterized level set blade topology optimization method based on conformal geometry theory and isogeometric analysis. Aiming at the technical challenges of difficult modeling and high cost of solving complex curved surfaces such as blades, the curved blade structure is mapped into a two-dimensional space, an orthogonal grid is established, and then isogeometric analysis is performed. The obtained model and the blade model subjected to mechanical analysis of fluid-solid coupling are used to obtain the load conditions, and the blade structure design is performed based on the level set method combined with conformal geometry and isogeometric analysis. Finally, the two-dimensional optimization result is inversely mapped back to the three-dimensional curved surface structure. This method can be directly used on blade structures. Similar structures include the front curved surface of aerospace rockets, submarine structures, wind blade structures, and various blade structures.

[0044] (2) The method of the present invention realizes dimensionality reduction of three-dimensional surfaces through conformal mapping theory and isogeometric level set method, which greatly reduces the computational cost and improves the optimization efficiency. At the same time, it combines the high-precision computational advantages of isogeometric analysis and the smooth structure of level set optimization to avoid the checkerboard problem of similar variable density methods, thus achieving the purpose of efficient and accurate optimization of surface structures. BRIEF DESCRIPTION OF THE DRAWINGS

[0045] Figure 1 is a flow chart of a parameterized level set blade topology optimization method based on conformal geometry theory and isogeometric analysis according to the present invention;

[0046] Figure 2 is a schematic diagram of the conformal mapping principle according to the method of the present invention;

[0047] Figure 3 is a barycentric interpolation coordinate diagram in the method according to the present invention;

[0048] Figure 4is the parameter domain Ω in the method according to the present invention 0 Relationship diagram with model Ω;

[0049] Figure 5 Schematic diagram of control points of medium geometry analysis grid according to the method of the present invention. DETAILED DESCRIPTION

[0050] The specific implementation modes of the present invention are described in detail below in conjunction with the accompanying drawings, but it should be understood that the protection scope of the present invention is not limited by the specific implementation modes.

[0051] Unless explicitly stated otherwise, throughout the specification and claims, the term “comprise” or variations such as “include” or “comprising”, etc., will be understood to include the stated elements or components but not to exclude other elements or components.

[0052] like Figures 1 to 5 As shown, according to a specific embodiment of the present invention, a parameterized level set blade topology optimization method based on conformal geometry theory and isogeometric analysis is used to perform lightweight design of blades while maintaining the original mechanical properties.

[0053] As a key component of turbines, wind turbines and other equipment, the lightweight design of blades directly affects the working efficiency of the equipment and the reliability of the design results. Blades are typical curved structures. Traditional optimization design is time-consuming and labor-intensive. Using conformal geometry theory to reduce dimensions can greatly reduce the computational cost. At the same time, isogeometric analysis methods have a strong solution accuracy in structural optimization.

[0054] A parametric level set blade topology optimization method based on conformal geometry theory and isogeometric analysis can be extended to the optimization design of other curved surface structures. The existing mixed flow blade is used as the design object, and it is input into the mesh software MeshLab, and the Delaunay triangulation is completed based on the Butterfly Subdivision algorithm. The following steps are included:

[0055] Step 1: Establish an initial blade model based on the existing blade shape;

[0056] According to the design requirements, the design variables and constraints are determined to construct the objective function and establish the mathematical model of the blade airfoil:

[0057]

[0058] Where is the Heaviside function, is the objective function, E is the elasticity tensor, ε is the strain, u is the displacement, v is the virtual displacement belonging to space u, and the inequality V(Ω) represents the volume constraint.

[0059] Step 2: Perform fluid-solid coupling mechanical analysis on the blade structure model to calculate the blade surface load;

[0060] The blade model was imported into Fluent. After blade meshing, boundary definition of inlet flow and velocity vector setting, the three-dimensional pressure of the fluid domain where the blade was located was calculated by Matlab fluid-solid coupling analysis.

[0061] Step 3: The blade structure model is conformally mapped into a two-dimensional rectangular domain, and a standard orthogonal grid with the same specifications as the mapped grid is established;

[0062] Using the Teichmüller extreme value mapping theory and calculation algorithm on the point cloud in the conformal mapping theory, the three-dimensional blade model is mapped into a triangulation of a two-dimensional rectangular domain, completing the mapping of the Riemann surface to the Euclidean plane, that is, the local scaling-conformal factor. On the basis of the triangulation, the barycentric interpolation is performed to establish a standard orthogonal grid equal to the domain mapping grid. Figure 2 As shown and Figure 3 shown.

[0063] Map the three-dimensional blade structure to the unit open disk D. For the mapping f:M→D, M and f can be calculated by solving the following Laplace equation:

[0064]

[0065] The Riemann metric is given by the arc length parameterization, defined as g = e 2ω(u,v) (dx 0 2 +dy 0 2 ), where (x 0 ,y 0 ) is the isothermal coordinate; the smooth mapping f:M→N is a scaling transformation, which is determined by the conformal factor scalar function e 2ω The local scaling operation of the control is performed, and then the mapped two-dimensional grid is established as an orthogonal grid.

[0066] Step 4: Establish an optimized mathematical model of the blade structure;

[0067] According to the surface load and volume constraints calculated by Fluent, the minimum compliance of the objective function is constructed, and the optimization mathematical model of the blade is established.

[0068] Step 5: Isogeometric analysis of blade structure; including:

[0069] On the basis of the orthogonal grid, control points and weights are designed, and the B-spline surface is defined by basis functions and control points, as shown in equation (3) and Figure 4 As shown and Figure 5 As shown. The B-spline surface is defined as:

[0070]

[0071] Where N i,p (u), N j,p (v) is the p-order B-spline basis function defined on the knot vectors U and V, controlling the fixed point P i,j .

[0072] Step 6: Parameterized level set optimization;

[0073] The parameterized level set function of isogeometric analysis is introduced, and combined with conformal bonding theory, the governing equations and the modified Hamilton-Jacobi equations are derived, and then the level set function is evolved;

[0074] According to the parameterized level set method

[0075]

[0076] Introducing parameterized level set functions for isogeometric analysis

[0077]

[0078] where φ i (t) is the expansion coefficient associated with the ith grid point, N i (x) is the corresponding basis function;

[0079] According to formulas (4) and (5), the Hamilton-Jacobi equation can be updated as

[0080]

[0081] Combining the conformal geometry theory, the modified isogeometry-conformal geometry-level set function is expressed as follows according to formula (2):

[0082]

[0083] Among them, v is related to the time derivative of the expansion coefficient n It is expressed as:

[0084]

[0085] According to the sensitivity analysis, the normal velocity field is constructed, and the level set function is evolved by solving the Hamilton-Jacobi equation in two-dimensional space.

[0086] Step 7: Inversely map the blade structure to complete the optimization process;

[0087] After completing all optimization cycles based on the above steps, a two-dimensional optimized structure is obtained. After the two-dimensional mesh is inversely mapped to the three-dimensional blade surface, i.e., the blade, through barycentric interpolation, the optimization process is completed.

[0088] The foregoing description of specific exemplary embodiments of the present invention is for the purpose of illustration and demonstration. These descriptions are not intended to limit the present invention to the precise form disclosed, and it is clear that many changes and variations can be made based on the above teachings. The purpose of selecting and describing the exemplary embodiments is to explain the specific principles of the present invention and its practical application, so that those skilled in the art can realize and utilize various different exemplary embodiments of the present invention and various different selections and changes. The scope of the present invention is intended to be limited by the claims and their equivalents.

Claims

1. A parameterized level set blade topology optimization method based on conformal geometry theory and isogeometric analysis, characterized in that: The following steps are involved: Step 1: Establish an initial blade model based on the existing blade shape; Step 2: Perform fluid-solid coupling mechanical analysis on the blade structure model to calculate the blade surface load; Step 3: The blade structure model is conformally mapped into a two-dimensional rectangular domain, and a standard orthogonal grid with the same specifications as the mapped grid is established; Step 4: Establish an optimized mathematical model of the blade structure; Step 5: Isogeometric analysis of blade structure; Step 6: Parameterized level set optimization; Step 7: Inverse map the blade structure to complete the optimization process.

2. The parameterized level set blade topology optimization method based on conformal geometry theory and isogeometric analysis according to claim 1 is characterized in that: Step 1: Establishing the initial blade model includes: determining the design variables and constraints according to the design requirements to construct the objective function and establish the blade airfoil mathematical model: Where is the Heaviside function, is the objective function, E is the elasticity tensor, ε is the strain, u is the displacement, v is the virtual displacement belonging to space u, and the inequality V(Ω) represents the volume constraint.

3. The parameterized level set blade topology optimization method based on conformal geometry theory and isogeometric analysis according to claim 1 is characterized in that: Step 2 is to perform fluid-solid coupling mechanical analysis on the blade structure model, including: dividing the blade structure model through blade meshing, setting the boundary definition of the inlet flow and velocity vector, and calculating the three-dimensional pressure of the fluid domain where the blade is located through fluid-solid coupling analysis.

4. The parameterized level set blade topology optimization method based on conformal geometry theory and isogeometric analysis according to claim 1 is characterized in that: Step three establishes a standard orthogonal grid with the same specifications as the mapping grid, including: mapping the three-dimensional blade model into a triangulated decomposition of a two-dimensional rectangular domain, completing the mapping of the Riemann surface to the Euclidean plane, that is, the local scaling-conformal factor, and performing barycentric interpolation on the basis of the triangulated decomposition to establish a standard orthogonal grid with the same specifications as the domain mapping grid.

5. The parameterized level set blade topology optimization method based on conformal geometry theory and isogeometric analysis according to claim 4 is characterized in that: Step 3: Create a standard orthogonal grid with the same specifications as the mapping grid, including: Mapping the 3D leaf structure to the unit open disk For a map f: M and f in can be calculated by solving the following Laplace equation: The Riemann metric is given by the arc length parameterization, defined as g = e 2ω(u,v) (dx0 2 +dy0 2 ), where (x0, y0) are isothermal coordinates; the smooth mapping f:M→N is a scaling transformation, which is determined by the conformal factor scalar function e 2ω The local scaling operation of the control is performed, and then the mapped two-dimensional grid is established into an orthogonal grid.

6. The parameterized level set blade topology optimization method based on conformal geometry theory and isogeometric analysis according to claim 1 is characterized in that: Step 4: Establishing the optimized mathematical model of the blade structure includes: constructing the minimum compliance of the objective function according to the calculated blade surface load and volume constraint conditions, and establishing the optimized mathematical model of the blade.

7. The parameterized level set blade topology optimization method based on conformal geometry theory and isogeometric analysis according to claim 1 is characterized in that: Step 5: Isogeometric analysis of the blade structure, including: On the basis of the orthogonal grid, control points and weights are designed, and the B-spline surface is defined by basis functions and control points. The B-spline surface is defined as: Where N i,p (u), N j,p (v) is the p-order B-spline basis function defined on the knot vectors U and V, controlling the fixed point P i,j .

8. The parameterized level set blade topology optimization method based on conformal geometry theory and isogeometric analysis according to claim 1 is characterized in that: Step 6: Parameterized level set optimization, including: introducing the parameterized level set function of isogeometric analysis, combining with conformal bonding theory, deriving the control equation, the modified Hamilton-Jacobi equation, and then evolving the level set function; According to the parameterized level set method Introducing parameterized level set functions for isogeometric analysis where φ i (t) is the expansion coefficient associated with the ith grid point, N i (x) is the corresponding basis function; According to formulas (4) and (5), the Hamilton-Jacobi equation can be updated as Combining the conformal geometry theory, the modified isogeometry-conformal geometry-level set function is expressed as follows according to formula (2): Among them, v is related to the time derivative of the expansion coefficient n It is expressed as: According to the sensitivity analysis, the normal velocity field is constructed, and the level set function is evolved by solving the Hamilton-Jacobi equation in two-dimensional space.

9. The parameterized level set blade topology optimization method based on conformal geometry theory and isogeometric analysis according to claim 1, characterized in that: Step seven is to inversely map the blade structure, including: after completing the optimization cycle, a two-dimensional optimized structure is obtained, and then the two-dimensional grid is inversely mapped to the three-dimensional blade surface through barycentric interpolation to complete the optimization process.

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